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          <link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetIncomeLossAvailableToCommonStockholdersDilutedAbstract" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersDilutedAbstract"/>
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          <link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityEmergingGrowthCompany" xlink:label="dei_EntityEmergingGrowthCompany"/>
          <link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IndefiniteLivedIntangibleAssetsByMajorClassAxis" xlink:label="us-gaap_IndefiniteLivedIntangibleAssetsByMajorClassAxis"/>
          <link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AreaOfLand" xlink:label="us-gaap_AreaOfLand"/>
          <link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredGoodwillAndLiabilitiesAssumedNet" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredGoodwillAndLiabilitiesAssumedNet"/>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriodWeightedAverageGrantDateFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Equity Instruments Other than Options, Vested in Period, Weighted Average Grant Date Fair Value</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromSaleOfPropertyPlantAndEquipment_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Proceeds from Sale of Property, Plant, and Equipment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromSaleOfPropertyPlantAndEquipment_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Proceeds from Sale of Property, Plant, and Equipment, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromSaleOfPropertyPlantAndEquipment_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Proceeds from sale of property and equipment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SoftwareDevelopmentMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Software [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SoftwareDevelopmentMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Software Development [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AggregateOfNumberOfWarrantsToPurchase_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Aggregate of number of Warrants to purchase</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AggregateOfNumberOfWarrantsToPurchase_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Aggregate of number of Warrants to purchase</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AggregateOfNumberOfWarrantsToPurchase_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Aggregate of number of Warrants to purchase</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ConcentrationRiskByTypeAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Concentration Risk Type [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfRelatedPartyTransactionsTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule of Related Party Transactions [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfRelatedPartyTransactionsTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Related Party Transactions</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityIncorporationStateCountryCode_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Entity Incorporation, State or Country Code</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityIncorporationStateCountryCode_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Entity Incorporation, State or Country Code</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockSharesIssued_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Preferred Stock, Shares Issued</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockSharesIssued_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Preferred stock, shares issued (in shares)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInAccountsPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Increase (Decrease) in Accounts Payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInAccountsPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accounts payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInAccountsPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Reduction in accounts payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInAccountsPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Increase (Decrease) in Accounts Payable, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_RestatementAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Revision of Prior Period [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationContingentConsiderationArrangementsChangeInAmountOfContingentConsiderationLiability1_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Contingent Consideration, Change in Contingent Consideration, Liability, Increase (Decrease)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfDebtInstrumentsTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule of Long-Term Debt Instruments [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfDebtInstrumentsTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Company's outstanding debt</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PlanNameAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Plan Name [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NetWorkingCapitalAdjustmentFromBusinessCombinations_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Net working capital adjustment from business combinations</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NetWorkingCapitalAdjustmentFromBusinessCombinations_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Net Working Capital Adjustment From Business Combinations</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NetWorkingCapitalAdjustmentFromBusinessCombinations_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Net Working Capital Adjustment From Business Combinations</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityCurrentReportingStatus_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Entity Current Reporting Status</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityCurrentReportingStatus_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Entity Current Reporting Status</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DeemedSharesIssuedPricePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Deemed Shares Issued Price Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DeemedSharesIssuedPricePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Deemed Shares Issued Price Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DeemedSharesIssuedPricePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Deemed issue prices</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingExpensesAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Operating Expenses [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingExpensesAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Operating expenses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AverageValuePerUnitAsPerMonteCarloSimulations_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Average value per unit as per Monte Carlo simulations</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AverageValuePerUnitAsPerMonteCarloSimulations_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Average value per unit as per Monte Carlo simulations</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AverageValuePerUnitAsPerMonteCarloSimulations_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Average value per unit as per Monte Carlo simulations</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueInputsLevel2Member_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Value, Inputs, Level 2 [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueInputsLevel2Member_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Level 2</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Share-BasedCompensationArrangementsByShare-BasedPaymentAwardOptionsCanceledInPeriodWeightedAverageExercisePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangements by Share-Based Payment Award, Options, Canceled in Period, Weighted Average Exercise Price</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Share-BasedCompensationArrangementsByShare-BasedPaymentAwardOptionsCanceledInPeriodWeightedAverageExercisePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Share-Based Compensation Arrangements by Share-Based Payment Award, Options, Canceled in Period, Weighted Average Exercise Price</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Share-BasedCompensationArrangementsByShare-BasedPaymentAwardOptionsCanceledInPeriodWeightedAverageExercisePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Weighted-Average Exercise Price, Options canceled</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SMINoteMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">SMI Note</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SMINoteMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">SMI Note [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SMINoteMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">SMI Note</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommitmentsAndContingenciesDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Commitments and Contingencies Disclosure [Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommitmentsAndContingenciesDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Commitments and Contingencies</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DiscountForLackOfMarketability_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Discount For Lack Of Marketability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DiscountForLackOfMarketability_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Discount For Lack Of Marketability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DiscountForLackOfMarketability_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Discount for lack of marketability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonStockSharesAuthorized_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Common Stock, Shares Authorized</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonStockSharesAuthorized_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Common Stock, Shares Authorized</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfPreferredStockDividend_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Issuance of Preferred Stock Dividend</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfPreferredStockDividend_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Issuance of Preferred Stock Dividend</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfPreferredStockDividend_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Issuance of Preferred Stock Dividend</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_TemporaryEquityParOrStatedValuePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Temporary Equity, Par or Stated Value Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationIncurredCost_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Incurred Cost</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationIncurredCost_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination Incurred Cost</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationIncurredCost_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Business Combination Incurred Cost</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_WeightedAverageNumberBasicAndDilutedSharesOutstandingAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Weighted Average Number Basic And Diluted Shares Outstanding</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_WeightedAverageNumberBasicAndDilutedSharesOutstandingAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Weighted Average Number Basic And Diluted Shares Outstanding [Abstract]</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_ProductOrServiceAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Product and Service [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockVotingRights_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Preferred Stock, Voting Rights</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockVotingRights_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Preferred Stock Voting Rights</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_DocumentFiscalPeriodFocus_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Document Fiscal Period Focus</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RevenueFromContractWithCustomerAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Revenue from Contract with Customer [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SecuredOverFinancingRateMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Secured Over Financing Rate [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SecuredOverFinancingRateMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Secured Over Financing Rate [Member]</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ContractualObligation_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Contractual Obligation</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherAccruedLiabilitiesCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Other accrued expenses</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CashAndCashEquivalentsAtCarryingValue_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Cash and Cash Equivalent, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_WeightedAverageNumberOfSharesOutstandingDilutedDisclosureItemsAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Weighted Average Number of Shares Outstanding Reconciliation [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_WeightedAverageNumberOfSharesOutstandingDilutedDisclosureItemsAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Denominator:</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfMaturitiesOfLongTermDebtTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule of Maturities of Long-Term Debt [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfMaturitiesOfLongTermDebtTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Principal Payments under Term Loans</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NotesPayableCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Notes payable, current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NotesPayableCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Notes Payable, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NotesPayableCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Notes Payable, Current, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NotesPayableCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Notes payable, current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesJWarrantsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Series J Warrants [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesJWarrantsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Series J Warrants [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesJWarrantsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Series J Warrants</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonStockSharesIssued_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Common Stock, Shares, Issued</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonStockSharesIssued_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Common stock, shares, issued (in shares)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NumberOfShareSold_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Number Of Share Sold</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NumberOfShareSold_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Number Of Share Sold</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NumberOfShareSold_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Number Of Share Sold</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueAssetsMeasuredOnRecurringBasisUnobservableInputReconciliationByAssetClassDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Asset Class [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StatementTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Statement [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StatementTable_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Statement [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Rule10B51ArrModifiedFlag_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Rule 10b5 1 Arr Modified [Flag]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Rule10B51ArrModifiedFlag_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Rule 10b5-1 Arrangement Modified [Flag]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Rule10B51ArrModifiedFlag_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Rule 10b5-1 Arrangement Modified</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RestrictedCashNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Restricted Cash, Noncurrent</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RestrictedCashNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Restricted cash, less current portion</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PricePerSquareFoot_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Price per Square foot</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PricePerSquareFoot_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Price per Square foot</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PricePerSquareFoot_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Price per Square foot</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RightOfUseAssetObtainedInExchangeForOperatingLeaseLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Right-of-Use Asset Obtained in Exchange for Operating Lease Liability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RightOfUseAssetObtainedInExchangeForOperatingLeaseLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Right-of-use asset acquired in exchange for operating lease liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedArrangementsToObtainGoodsAndServicesAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Payment Arrangement, Disclosure [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsToAcquireBusinessesGross_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Payments to Acquire Businesses, Gross</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsToAcquireBusinessesGross_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Purchase price of acquisition</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StatementLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Statement [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StatementLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Statement [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionAcquireeDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationSeparatelyRecognizedTransactionsTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Net Impact of Acquisition</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationSeparatelyRecognizedTransactionsTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Separately Recognized Transaction [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_WarrantMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Warrant [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_WarrantMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Warrants</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityFilerCategory_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Entity Filer Category</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityFilerCategory_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Entity Filer Category</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedOperatingLeaseLiabilitiesNonCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Operating lease liabilities, non-current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedOperatingLeaseLiabilitiesNonCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination Recognized Identifiable Assets Acquired and Liabilities Assumed Operating Lease Liabilities Non Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedOperatingLeaseLiabilitiesNonCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Business Combination Recognized Identifiable Assets Acquired and Liabilities Assumed Operating Lease Liabilities Non Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LoansHeldForSaleMaturityDate_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Loan, Held-for-Sale, Maturity Date</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LoansHeldForSaleMaturityDate_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Term loan maturity date</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedIndefiniteLivedIntangibleAssets_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Intangible assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedIndefiniteLivedIntangibleAssets_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Recognized Asset Acquired, Identifiable Intangible Asset, Indefinite-Lived</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedIndefiniteLivedIntangibleAssets_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total acquired intangible assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LiabilitiesAndStockholdersEquity_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Liabilities and Equity</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LiabilitiesAndStockholdersEquity_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total liabilities, temporary equity, and stockholders' equity</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_VehiclesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Vehicles [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_VehiclesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Vehicles [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Debt Instrument [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Debt Instrument [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PayablesAndAccrualsAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Payables and Accruals [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInYearThree_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Long-Term Debt, Maturity, Year Three</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInYearThree_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">2028</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableWeightedAverageExercisePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Exercisable, Weighted Average Exercise Price</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableWeightedAverageExercisePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Weighted-Average Exercise Price Vested and exercisable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_UnauditedProFormaInformationMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Unaudited Pro Forma Information</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_UnauditedProFormaInformationMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Unaudited Pro Forma Information [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_UnauditedProFormaInformationMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Unaudited Pro Forma Information [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PropertyPlantAndEquipmentByTypeAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Long-Lived Tangible Asset [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PropertyPlantAndEquipmentByTypeAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Long-Lived Tangible Asset [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CommonStockWarrantsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Common Stock Warrants [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CommonStockWarrantsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Common Stock Warrants [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CommonStockWarrantsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Common Stock Warrants [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SciTecMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">SciTec [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SciTecMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">SciTec [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SciTecMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">SciTec</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CashAndCashEquivalentsFairValueDisclosure_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Fair value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CashAndCashEquivalentsFairValueDisclosure_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Cash and Cash Equivalents, Fair Value Disclosure</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ExtinguishmentOfDebtAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Extinguishment fees</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ExtinguishmentOfDebtAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Extinguishment of Debt, Amount</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SharesIssuedPricePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Shares Issued, Price Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SharesIssuedPricePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Price per share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SevenPointSeventyOnePercentDueTwoThousandTwentyNineMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Seven Point Seventy One Percent Due Two Thousand Twenty Nine [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SevenPointSeventyOnePercentDueTwoThousandTwentyNineMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Seven Point Seventy One Percent Due Two Thousand Twenty Nine [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SevenPointSeventyOnePercentDueTwoThousandTwentyNineMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">7.71% due 2029</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SubsequentEventLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Subsequent Event [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_Goodwill_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Goodwill, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_Goodwill_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Goodwill</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_Goodwill_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Goodwill</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_Goodwill_lbl" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel">Goodwill, Ending Balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_Goodwill_lbl" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel">Goodwill, Beginning Balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EffectiveIncomeTaxRateContinuingOperations_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Effective Income Tax Rate Reconciliation, Percent</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EffectiveIncomeTaxRateContinuingOperations_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Effective tax rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EffectiveIncomeTaxRateContinuingOperations_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Effective income tax rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LineOfCreditFacilityIncreaseDecreaseForPeriodNet_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Line of Credit Facility, Increase (Decrease), Net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LineOfCreditFacilityIncreaseDecreaseForPeriodNet_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Line of Credit Facility, Increase (Decrease), Net, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LineOfCreditFacilityIncreaseDecreaseForPeriodNet_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Line of Credit Facility, Increase</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseNextRollingTwelveMonths_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finite-Lived Intangible Assets, Amortization Expense, Next Rolling 12 Months</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseNextRollingTwelveMonths_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">2027</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NonCancelableSubscriptionBasedServicesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Non-Cancelable Subscription-Based Services</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NonCancelableSubscriptionBasedServicesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Non-Cancelable Subscription-Based Services [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NonCancelableSubscriptionBasedServicesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Non-Cancelable Subscription-Based Services</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNetAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Assets acquired and liabilities assumed</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNetAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Recognized Asset Acquired to Liability Assumed, Excess (Less) [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AccuredInterestFromSubordinatedConvertiblePromissoryNotes_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Accured Interest From Subordinated Convertible Promissory Notes</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AccuredInterestFromSubordinatedConvertiblePromissoryNotes_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Accured Interest From Subordinated Convertible Promissory Notes</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AccuredInterestFromSubordinatedConvertiblePromissoryNotes_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accured Interest From Subordinated Convertible Promissory Notes</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCapitalLeaseObligation_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Recognized Liability Assumed, Lease Obligation</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCapitalLeaseObligation_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Operating lease liabilities, current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesJWarrantMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Series J Warrant [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesJWarrantMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Series J Warrant [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesJWarrantMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Series J Warrant</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredFinanceCostsNet_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Debt Issuance Costs, Net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredFinanceCostsNet_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Debt Issuance Costs, Net, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredFinanceCostsNet_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Debt Issuance costs</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementByShare-BasedPaymentAwardEquityInstrumentsOtherThanOptionsCancelledInPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share Based Compensation Arrangement by Share-Based Payment Award, Equity Instruments Other Than Options, Cancelled in Period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementByShare-BasedPaymentAwardEquityInstrumentsOtherThanOptionsCancelledInPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Share Based Compensation Arrangement by Share-Based Payment Award, Equity Instruments Other Than Options, Cancelled in Period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementByShare-BasedPaymentAwardEquityInstrumentsOtherThanOptionsCancelledInPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Number of Options Outstanding, Options canceled</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TwoThousandTwentyFiveOmnibusIncentivePlanMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Two Thousand Twenty Five Omnibus Incentive Plan [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TwoThousandTwentyFiveOmnibusIncentivePlanMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Two Thousand Twenty Five Omnibus Incentive Plan [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TwoThousandTwentyFiveOmnibusIncentivePlanMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">2025 Omnibus Incentive Plan</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NetWorkingCapitalAdjustment_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Net working capital adjustment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NetWorkingCapitalAdjustment_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Net working capital adjustment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NetWorkingCapitalAdjustment_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Net working capital adjustment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RelatedPartyMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Related Party [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RelatedPartyMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Related Party</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ConversionOfStockAmountConverted1_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Conversion of Stock, Amount Converted</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ConversionOfStockAmountConverted1_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Conversion of Preferred Stock to common stock upon initial public offering</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PropertyPlantAndEquipmentLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Property, Plant and Equipment [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PropertyPlantAndEquipmentLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Property, Plant and Equipment [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NotesPayableAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Notes Payable [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetUsefulLife_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finite-Lived Intangible Asset, Useful Life</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetUsefulLife_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Useful Life</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_ShareRepurchaseProgramAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share Repurchase Program [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FireflyMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Firefly</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FireflyMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Firefly [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FireflyMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Firefly [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_VestingDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Vesting [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriodGross_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Grants in Period, Gross</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriodGross_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Number of Options Outstanding, Options granted</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DisaggregationOfRevenueTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Disaggregation of Revenue [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SaleLeasebackTransactionNameDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Sale Leaseback Transaction, Name [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfEarningsPerShareBasicAndDilutedTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule of Earnings Per Share, Basic and Diluted [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfEarningsPerShareBasicAndDilutedTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Computation of Basic and Diluted Net Loss Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_TrademarksAndTradeNamesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Trademarks and Trade Names [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_TrademarksAndTradeNamesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Trade name and trademarks</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LesseeFinanceLeaseDescriptionAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Lessee, Finance Lease, Description [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LesseeFinanceLeaseDescriptionAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Finance lease expense:</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromIssuanceOfPreferredStockAndPreferenceStock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Proceeds from Issuance of Preferred Stock and Preference Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromIssuanceOfPreferredStockAndPreferenceStock_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Proceeds from issuance of Preferred Stock, net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromIssuanceOfPreferredStockAndPreferenceStock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Proceeds from issuance of Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_ScheduleOfEquityMethodInvestmentEquityMethodInvesteeNameAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Investment, Name [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EarningsPerShareBasic_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Earnings Per Share, Basic</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EarningsPerShareBasic_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Net loss per share attributable to common stockholders - basic</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EarningsPerShareBasic_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Basic</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DisaggregationOfRevenueTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Disaggregation of Revenue [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DisaggregationOfRevenueTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Disaggregated Revenue</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Outstanding, Number</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel">Shares, Outstanding ending balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel">Shares, Outstanding beginning balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsVestedWeightedAverageGrantDateFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Vested, Weighted Average Grant Date Fair Value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsVestedWeightedAverageGrantDateFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Weighted-average grant date fair value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssued1_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Stock Issued</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssued1_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Initial fair value of warrants at issuance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SpecificAdditionalInterestRate_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Specific Additional Interest Rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SpecificAdditionalInterestRate_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Specific Additional Interest Rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SpecificAdditionalInterestRate_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Specific additional interest rate of premium</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RevolvingCreditFacilityMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Revolving Credit Facility [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RevolvingCreditFacilityMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Revolving Credit Facility</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_OutstandingBorrowingsOnRevolvingCreditFacilityMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Outstanding borrowings on Revolving Credit Facility [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_OutstandingBorrowingsOnRevolvingCreditFacilityMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Outstanding borrowings on Revolving Credit Facility [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_OutstandingBorrowingsOnRevolvingCreditFacilityMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Outstanding borrowings on Revolving Credit Facility</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LiabilitiesCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Liabilities, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LiabilitiesCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total current liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInAccountsReceivable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Increase (Decrease) in Accounts Receivable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInAccountsReceivable_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel">Accounts receivable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInAccountsReceivable_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Accounts receivable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Adjustment to Reconcile Net Income to Cash Provided by (Used in) Operating Activity [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Adjustments to reconcile net loss to net cash used in operating activities:</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommodityContractAssetCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Commodity Contract Asset, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommodityContractAssetCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Total contract assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommodityContractAssetCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total contract assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EntityWideRevenueMajorCustomerLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Revenue, Major Customer [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExercisePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Fair Value Assumptions, Exercise Price</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExercisePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Contractual purchase price for Series J Preferred Stock (per share)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PaymentsOnNotesPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Payments on Notes Payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PaymentsOnNotesPayable_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Payments on Notes Payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PaymentsOnNotesPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Payments on Notes Payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PaymentsOnNotesPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Payments on Notes Payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OptionIndexedToIssuersEquityShares_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Option Indexed to Issuer's Equity, Shares</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OptionIndexedToIssuersEquityShares_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Number of the Shares Underlying the RPM Call Option</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_ShareRepurchaseProgramDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share Repurchase Program [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_InterestExpenseContractual_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Interest Expense Contractual</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_InterestExpenseContractual_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Interest Expense Contractual</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_InterestExpenseContractual_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Interest Expense Contractual</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_WeightedAverageFairValuePerWarrant_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Weighted Average Fair Value Per Warrant</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_WeightedAverageFairValuePerWarrant_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Weighted Average Fair Value Per Warrant</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_WeightedAverageFairValuePerWarrant_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Weighted average value per Series J Warrant</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_WeightedAveragePricePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Weighted Average Price Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_WeightedAveragePricePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Weighted Average Price Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_WeightedAveragePricePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Weighted average Series D Preferred Stock Price</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsOfDebtIssuanceCosts_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Payments of Debt Issuance Costs</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsOfDebtIssuanceCosts_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel">Costs associated with the credit facility</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsOfDebtIssuanceCosts_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Debt issuance costs</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsOfDebtIssuanceCosts_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Payments of debt issuance costs</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_ConsolidationItemsDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Consolidation Items [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_ConsolidationItemsDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Consolidation Items [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockholdersEquityAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Equity, Attributable to Parent [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockholdersEquityAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Stockholders' equity</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntitySmallBusiness_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Entity Small Business</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntitySmallBusiness_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Entity Small Business</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ProceedsFromRepaymentOfEmployeeNote_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Proceeds From Repayment Of Employee Note</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ProceedsFromRepaymentOfEmployeeNote_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Proceeds From Repayment Of Employee Note</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ProceedsFromRepaymentOfEmployeeNote_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Proceeds from repayment of employee note</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInAccruedLiabilities_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Increase (Decrease) in Accrued Liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInAccruedLiabilities_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accrued expenses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RetainedEarningsAccumulatedDeficit_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Retained Earnings (Accumulated Deficit)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RetainedEarningsAccumulatedDeficit_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accumulated deficit</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RetainedEarningsAccumulatedDeficit_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Retained Earnings (Accumulated Deficit), Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IncreaseDecreaseAccountsPayableRelatedParties_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Increase Decrease Accounts Payable Related Parties</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IncreaseDecreaseAccountsPayableRelatedParties_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Increase Decrease Accounts Payable Related Parties</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IncreaseDecreaseAccountsPayableRelatedParties_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accounts payable and accrued expenses - related parties</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherCommitmentsLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Other Commitments [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherCommitmentsLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Other Commitments [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccretionExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accretion expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccretionExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Accretion Expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Redeemable Convertible Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromIssuanceInitialPublicOffering_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Proceeds from Issuance Initial Public Offering</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromIssuanceInitialPublicOffering_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Proceeds from initial public offering</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromIssuanceInitialPublicOffering_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Aggregate Purchased price</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SaleOfStockPricePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Sale of Stock, Price Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SaleOfStockPricePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Sale of Stock, Price Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisesInPeriodTotalIntrinsicValue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Exercises in Period, Intrinsic Value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisesInPeriodTotalIntrinsicValue_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Aggregate intrinsic value of the options exercisable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DisclosureOfCompensationRelatedCostsShareBasedPaymentsTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Payment Arrangement [Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DisclosureOfCompensationRelatedCostsShareBasedPaymentsTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Stock-Based Compensation</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PercentageOfNonAccruingOrignalIssuePricePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Percentage Of Non Accruing Orignal Issue Price Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PercentageOfNonAccruingOrignalIssuePricePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Percentage Of Non Accruing Orignal Issue Price Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PercentageOfNonAccruingOrignalIssuePricePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Percentage of Non Accruing Deemed Issue Price Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementOptionFairValueOutstandingNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Number of units subject to purchase by majority sponsor</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementOptionFairValueOutstandingNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share Based Compensation Arrangement Option Fair Value Outstanding Number</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementOptionFairValueOutstandingNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Share Based Compensation Arrangement Option Fair Value Outstanding Number</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Organization, Consolidation and Presentation of Financial Statements [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LeasesAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Leases [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonClassAMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Common Class A [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonClassAMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Common Class A</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ContractWithCustomerLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Contract with Customer, Liability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ContractWithCustomerLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Contract liabilities (current and non-current)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ContractWithCustomerLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Contract with Customer, Liability, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LesseeLeaseDescriptionLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Lessee, Lease, Description [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_AmendmentFlag_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Amendment Flag</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_AmendmentFlag_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Amendment Flag</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_LeaseRemainingLeaseTerm_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Lease Remaining Lease Term</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_LeaseRemainingLeaseTerm_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Lease Remaining Lease Term</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_LeaseRemainingLeaseTerm_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Remaining lease term</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_ProductsAndServicesDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Product and Service [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByWarrantsOrRights_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Warrant issued</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByWarrantsOrRights_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Class of Warrant or Right, Number of Securities Called by Warrants or Rights</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FinanceLeaseInterestExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finance Lease, Interest Expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FinanceLeaseInterestExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Interest on lease liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LiabilitiesAndStockholdersEquityAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Liabilities and Equity [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LiabilitiesAndStockholdersEquityAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Liabilities, temporary equity, and stockholders' deficit</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityAddressPostalZipCode_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Entity Address, Postal Zip Code</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityAddressPostalZipCode_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Entity Address, Postal Zip Code</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivities_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Cash Provided by (Used in) Operating Activity, Including Discontinued Operation</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivities_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Net cash used in operating activities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivities_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Cash flows from operations</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivities_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Cash flows from operations</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivities_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel">Cash flows from operations</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInOperatingLeaseLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Increase (Decrease) in Operating Lease Liability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInOperatingLeaseLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Lease liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Non-CashInterestExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Non-Cash Interest Expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Non-CashInterestExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Non-Cash Interest Expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Non-CashInterestExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Non-cash interest expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionProFormaEarningsPerShareBasic_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Pro Forma Information, Pro Forma Earnings Per Share, Basic</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionProFormaEarningsPerShareBasic_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Pro forma net loss per common share, Basic</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NonRule10B51ArrModifiedFlag_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Non Rule 10b5 1Arr Modified [Flag]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NonRule10B51ArrModifiedFlag_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Non-Rule 10b5-1 Arrangement Modified [Flag]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NonRule10B51ArrModifiedFlag_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Non-Rule 10b5-1 Arrangement Modified</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Common Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Common&#160;Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementOptionFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share Based Compensation Arrangement Option Fair Value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementOptionFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Share Based Compensation Arrangement Option Fair Value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementOptionFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Value of forward obligation (per share)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedInventory_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Recognized Asset Acquired, Inventory, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedInventory_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Inventory</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInNextTwelveMonths_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Long-Term Debt, Maturity, Year One</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInNextTwelveMonths_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">2026</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsGrantsInPeriodWeightedAverageExercisePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangements by Share-Based Payment Award, Options, Grants in Period, Weighted Average Exercise Price</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsGrantsInPeriodWeightedAverageExercisePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Weighted-Average Exercise Price, Options granted</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccumulatedDepreciationDepletionAndAmortizationPropertyPlantAndEquipment_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Accumulated Depreciation, Depletion and Amortization, Property, Plant, and Equipment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccumulatedDepreciationDepletionAndAmortizationPropertyPlantAndEquipment_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel">Less: Accumulated depreciation</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BelcanLLCMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Belcan, LLC [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BelcanLLCMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Belcan, LLC [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BelcanLLCMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Belcan, LLC</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentInterestRateEffectivePercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Debt Instrument, Interest Rate, Effective Percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentInterestRateEffectivePercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Effective interest rate of the term loans</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementOptionOutstandingFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Value of RPM Option</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementOptionOutstandingFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share Based Compensation Arrangement Option Outstanding Fair Value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementOptionOutstandingFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Share Based Compensation Arrangement Option Outstanding Fair Value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityAddressAddressLine1_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Entity Address, Address Line One</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityAddressAddressLine1_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Entity Address, Address Line One</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfShareBasedPaymentAwardStockOptionsValuationAssumptionsTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Summary of assumptions fair value of options granted</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfShareBasedPaymentAwardStockOptionsValuationAssumptionsTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule of Share-Based Payment Award, Stock Options, Valuation Assumptions [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TwoThousandSeventeenEquityIncentivePlanMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Two Thousand Seventeen Equity Incentive Plan [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TwoThousandSeventeenEquityIncentivePlanMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Two Thousand Seventeen Equity Incentive Plan [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TwoThousandSeventeenEquityIncentivePlanMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">2017 Equity Incentive Plans</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueByFairValueHierarchyLevelAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Value Hierarchy and NAV [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EarningsPerShareAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Earnings Per Share [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EarningsPerShareAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Net loss per common share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_TemporaryEquityCarryingAmountAttributableToParent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Temporary Equity, Carrying Amount, Attributable to Parent</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_TemporaryEquityCarryingAmountAttributableToParent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Temporary equity, Carrying value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SupplementalCashFlowInformationAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Supplemental Cash Flow Information [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SupplementalCashFlowInformationAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Supplemental disclosures of cash flow information</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TwoThousandTwentyFiveEmployeeStockPurchasePlanMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">2025 Employee Stock Purchase Plan</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TwoThousandTwentyFiveEmployeeStockPurchasePlanMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Two Thousand Twenty Five Employee Stock Purchase Plan [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TwoThousandTwentyFiveEmployeeStockPurchasePlanMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Two Thousand Twenty Five Employee Stock Purchase Plan [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GoodwillAndIntangibleAssetsDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Goodwill and Intangible Assets Disclosure [Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GoodwillAndIntangibleAssetsDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Goodwill</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingLeaseRightOfUseAsset_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Operating Lease, Right-of-Use Asset</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingLeaseRightOfUseAsset_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Right-of-use assets - operating leases</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingLeaseRightOfUseAsset_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Operating lease right-of-use assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IncreaseDecreaseInAdvancedPayments_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Advanced payments</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IncreaseDecreaseInAdvancedPayments_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Increase (Decrease) In Advanced Payments</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IncreaseDecreaseInAdvancedPayments_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Increase (Decrease) In Advanced Payments</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TermLoanMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Term Loan</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TermLoanMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Term Loan [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TermLoanMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Term Loan</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentAssetsOther_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Recognized Asset Acquired, Other Asset, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentAssetsOther_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Other current assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_DocumentType_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Document Type</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_DocumentType_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Document Type</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GrossProfit_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Gross Profit</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GrossProfit_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Gross profit</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_TypeOfArrangementAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Collaborative Arrangement and Arrangement Other than Collaborative [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromCommissionsReceived_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Proceeds from Commissions Received</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromCommissionsReceived_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Proceeds of commission received</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ArrangementsAndNonarrangementTransactionsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Collaborative Arrangement and Arrangement Other than Collaborative [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityCentralIndexKey_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Entity Central Index Key</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityCentralIndexKey_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Central Index Key</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LineOfCreditFacilityTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Line of Credit Facility [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FollowOnOpportunitiesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Follow on opportunities[Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FollowOnOpportunitiesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Follow on opportunities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FollowOnOpportunitiesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Follow-on opportunities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ValuationTechniqueAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Valuation Approach and Technique [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Cash, Cash Equivalent, Restricted Cash, and Restricted Cash Equivalent, Period Increase (Decrease), Including Exchange Rate Effect and Discontinued Operation</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Net (decrease) increase in cash and cash equivalents and restricted cash</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SubsequentEventTypeDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Subsequent Event Type [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueAdjustmentOfWarrants_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Value Adjustment of Warrants</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueAdjustmentOfWarrants_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Change in fair value of warrant liability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueAdjustmentOfWarrants_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Change in fair value of warrant liability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_InterestIncomeOperating_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Interest Income, Operating, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_InterestIncomeOperating_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Interest income</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_InterestIncomeOperating_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Interest Income</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Antidilutive Securities Excluded from Computation of Earnings Per Share, Amount</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredCompensationArrangementWithIndividualCommonStockReservedForFutureIssuance_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Deferred Compensation Arrangement with Individual, Common Stock Reserved for Future Issuance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredCompensationArrangementWithIndividualCommonStockReservedForFutureIssuance_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Common stock available for issuance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CustomerOneMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Customer One [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CustomerOneMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Customer One Member</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CustomerOneMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Customer 1</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractorsAndOutsideServices_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Contractors and Outside Services</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractorsAndOutsideServices_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Contractors and Outside Services</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractorsAndOutsideServices_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Contractors and outside services</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractorsAndOutsideServices_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Contractors and outside services</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AdjustmentsToAdditionalPaidInCapitalShareBasedCompensationStockOptionsRequisiteServicePeriodRecognition_lbl" xlink:role="http://www.xbrl.org/2003/role/label">APIC, Share-Based Payment Arrangement, Option, Increase for Cost Recognition</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AdjustmentsToAdditionalPaidInCapitalShareBasedCompensationStockOptionsRequisiteServicePeriodRecognition_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Stock-based compensation</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonStockParOrStatedValuePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Common Stock, Par or Stated Value Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonStockParOrStatedValuePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Common stock, par or stated value per share (in USD per share)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonStockParOrStatedValuePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Common stock retained par value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedDeferredTaxAssets_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Deferred project costs</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedDeferredTaxAssets_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Recognized Asset Acquired, Deferred Tax Asset</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingCostsAndExpenses_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Operating Costs and Expenses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingCostsAndExpenses_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Operating Costs and Expenses, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingCostsAndExpenses_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Expenses- related parties</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PreferredStockIssuedAtAPurchasePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Preferred Stock Issued at a Purchase Price</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PreferredStockIssuedAtAPurchasePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Preferred Stock Issued at a Purchase Price</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PreferredStockIssuedAtAPurchasePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Preferred stock issued at a purchase price</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IPOScenarioMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">IPO Scenario [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IPOScenarioMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">IPO Scenario [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IPOScenarioMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">IPO Scenario</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ExitFeePercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Exit Fee Percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ExitFeePercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Exit Fee Percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ExitFeePercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Exit fee</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ConvertibleSeriesSeed-1PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Convertible Series Seed-1 Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ConvertibleSeriesSeed-1PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Convertible Series Seed-1 Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ConvertibleSeriesSeed-1PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Convertible Series Seed-1 Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LiabilitiesAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Liabilities [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LiabilitiesAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Liabilities:</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BaseRateMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Base Rate [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BaseRateMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Base rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsOutstandingWeightedAverageRemainingContractualTerm2_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Weighted-Average Remaining Contractual Term, Outstanding</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsOutstandingWeightedAverageRemainingContractualTerm2_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Outstanding, Weighted Average Remaining Contractual Term</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInOtherOperatingLiabilities_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Increase (Decrease) in Other Operating Liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInOtherOperatingLiabilities_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Other liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ChangeInRight-Of-UseAssets_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Change in Right-Of-Use Assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ChangeInRight-Of-UseAssets_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Change in Right-Of-Use Assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ChangeInRight-Of-UseAssets_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Right-of-use assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CallOptionMember_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Call Option</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CallOptionMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Call Option [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractualObligationCancellationFeePercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Contractual Obligation, Cancellation Fee Percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractualObligationCancellationFeePercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Contractual Obligation, Cancellation Fee Percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractualObligationCancellationFeePercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Contractual obligation, cancellation fee</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssuedDuringPeriodSharesStockOptionsExercised_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Exercises in Period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssuedDuringPeriodSharesStockOptionsExercised_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Common stock, share exercised</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssuedDuringPeriodSharesStockOptionsExercised_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Number of Options Outstanding, Options exercised</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssuedDuringPeriodSharesStockOptionsExercised_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Number of Options Outstanding, Options exercised</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ITRelatedEquipmentMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">IT Related Equipment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ITRelatedEquipmentMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">IT Related Equipment [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ITRelatedEquipmentMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">IT related equipment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BlackScholesMertonModelMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Valuation Technique, Black-Scholes-Merton Model [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BlackScholesMertonModelMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Black-Scholes option-pricing valuation</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RepaymentsOfRelatedPartyDebt_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Repayments of Related Party Debt</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RepaymentsOfRelatedPartyDebt_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Repayments of debt</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TransactionRelatedExpenses_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Transaction-related expenses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TransactionRelatedExpenses_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Transaction Related Expenses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TransactionRelatedExpenses_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Transaction Related Expenses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccruedLiabilitiesCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Accrued Liabilities, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccruedLiabilitiesCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accrued expenses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccruedLiabilitiesCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Accrued expenses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_EightPointThirtyOnePercentDueTwoThousandThirtyOneMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Eight Point Thirty One Percent Due Two Thousand Thirty One [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_EightPointThirtyOnePercentDueTwoThousandThirtyOneMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Eight Point Thirty One Percent Due Two Thousand Thirty One [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_EightPointThirtyOnePercentDueTwoThousandThirtyOneMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">8.31% due 2031</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_CurrentFiscalYearEndDate_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Current Fiscal Year End Date</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_CurrentFiscalYearEndDate_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Current Fiscal Year End Date</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule of Long-Term Debt Instruments [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentTable_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Long-Term Debt Instruments [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CapitalizedComputerSoftwareAdditions_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Capitalized Computer Software, Additions</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CapitalizedComputerSoftwareAdditions_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Capitalized of software development</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ChangeOfControlScenarioMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Change of Control Scenario [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ChangeOfControlScenarioMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Change of Control Scenario [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ChangeOfControlScenarioMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Change of Control Scenario</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_DocumentQuarterlyReport_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Document Quarterly Report</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_DocumentQuarterlyReport_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Document Quarterly Report</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CustomerRelationshipsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Customer Relationships [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CustomerRelationshipsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Customer Relationships</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueByBalanceSheetGroupingTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Value, by Balance Sheet Grouping [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityAddressStateOrProvince_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Entity Address, State or Province</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityAddressStateOrProvince_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Entity Address, State or Province</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MajoritySponsorMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Majority Sponsor [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MajoritySponsorMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Majority Sponsor [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MajoritySponsorMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Majority Sponsor Top-Up</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsNonvestedNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Equity Instruments Other than Options, Nonvested, Number</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsNonvestedNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel">Number of shares, Ending Balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsNonvestedNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel">Number of shares, Beginning Balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsNonvestedNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Number of Options Outstanding, Unvested</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_EquityMethodInvesteeNameDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Investment, Name [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockSharesOutstanding_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Preferred Stock, Shares Outstanding</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockSharesOutstanding_lbl" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel">Preferred stock ending balance (in shares)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockSharesOutstanding_lbl" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel">Preferred stock beginning balance (in shares)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockSharesOutstanding_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Preferred stock, shares outstanding (in shares)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonStockCapitalSharesReservedForFutureIssuance_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Number of reserved shares</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonStockCapitalSharesReservedForFutureIssuance_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Common Stock, Capital Shares Reserved for Future Issuance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonStockCapitalSharesReservedForFutureIssuance_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Common stock available for issuance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ContractWithCustomerLiabilityRevenueRecognized_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Contract with Customer, Liability, Revenue Recognized</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ContractWithCustomerLiabilityRevenueRecognized_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Contract liability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ContractWithCustomerLiabilityRevenueRecognized_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Recognize Revenue</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionCostOfAcquiredEntityTransactionCosts_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Transaction Cost, Excluding Separately Recognized Transaction</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionCostOfAcquiredEntityTransactionCosts_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Transaction Cost</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ConversionOfStockSharesConverted1_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Conversion of Preferred Stock to common stock, Shares</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ConversionOfStockSharesConverted1_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Conversion of Stock, Shares Converted</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MinimumLiquidityMaintenanceCovenant_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Minimum Liquidity Maintenance Covenant</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MinimumLiquidityMaintenanceCovenant_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Minimum Liquidity Maintenance Covenant</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MinimumLiquidityMaintenanceCovenant_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Minimum liquidity</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccountsReceivableMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accounts Receivable [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccountsReceivableMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Accounts Receivable [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesMPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Series M Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesMPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Series M Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesMPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Series M Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesMPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Convertible Series M Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssuedDuringPeriodValueStockOptionsExercised_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Stock Issued During Period, Value, Stock Options Exercised</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssuedDuringPeriodValueStockOptionsExercised_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Options Exercised</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssuedDuringPeriodValueStockOptionsExercised_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Call Option derecognized</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfRevenueByMajorCustomersByReportingSegmentsTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Concentration of Credit Risk</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfRevenueByMajorCustomersByReportingSegmentsTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule of Revenue by Major Customers by Reporting Segments [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentUnamortizedDiscountPremiumNet_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Debt Instrument, Unamortized Discount (Premium), Net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentUnamortizedDiscountPremiumNet_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Debt Instrument, Unamortized Discount (Premium), Net, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentUnamortizedDiscountPremiumNet_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Less: Unamortized issuance costs</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentUnamortizedDiscountPremiumNet_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Unamortized issuance costs</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockOptionMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Equity Option [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockOptionMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Equity Option [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SubsequentEventMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Subsequent Event [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SubsequentEventMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Subsequent Event</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SwinglineLoans_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Sublimit for swingline loans</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SwinglineLoans_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Swingline Loans</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SwinglineLoans_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Swingline Loans</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredRevenueCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Deferred revenue, current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredRevenueCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Deferred Revenue, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredRevenueCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Deferred Revenue, Current, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PropertyPlantAndEquipmentAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Property, Plant and Equipment [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherAssetsNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Other Assets, Noncurrent</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherAssetsNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Other assets, less current portion</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsAvailableForGrantExercisedInPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share Based Compensation Arrangement by Share Based Payment Award, Options Available for Grant, Exercised in Period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsAvailableForGrantExercisedInPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Share Based Compensation Arrangement by Share Based Payment Award, Options Available for Grant, Exercised in Period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsAvailableForGrantExercisedInPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Options Available for Grant, Options exercised</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_GainOnSettlementOfContingentLiabilities_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Gain on settlement of contingent liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_GainOnSettlementOfContingentLiabilities_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Gain on settlement of contingent liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_GainOnSettlementOfContingentLiabilities_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Gain on settlement of contingent liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareByAntidilutiveSecuritiesAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Antidilutive Securities [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareByAntidilutiveSecuritiesAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Antidilutive Securities [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CustomerFourMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Customer Four [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CustomerFourMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Customer Four [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CustomerFourMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Customer 4</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentAssetsPrepaidExpenseAndOtherAssets_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Recognized Asset Acquired, Prepaid Expense and Other Asset, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentAssetsPrepaidExpenseAndOtherAssets_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Prepaid expenses and other current assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_WarrantExercisePriceDecrease_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Warrant, Exercise Price, Decrease</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_WarrantExercisePriceDecrease_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Warrant Exercise Price Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BasisOfPresentationAndSignificantAccountingPoliciesTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Basis of Presentation and Significant Accounting Policies [Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BasisOfPresentationAndSignificantAccountingPoliciesTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Summary of Significant Accounting Policies</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriodTotalFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Equity Instruments Other than Options, Vested in Period, Fair Value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriodTotalFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Fair value of RSUs vested</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_InterestExpenseDebt_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Interest Expense, Debt</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_InterestExpenseDebt_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total interest expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_InterestExpenseDebt_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Interest Expense Debt</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CustomerTwoMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Customer two Member</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CustomerTwoMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Customer Two [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CustomerTwoMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Customer 2</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherCommitmentsDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Other Commitments [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherCommitmentsDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Other Commitments [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_NameOfMajorCustomerDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Customer [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_NameOfMajorCustomerDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Customer [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsMajorClassNameDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finite-Lived Intangible Assets, Major Class Name [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CashMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Cash [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CashMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Cash</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CashProceeds_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Cash proceeds</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CashProceeds_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Cash Proceeds</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CashProceeds_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Cash Proceeds</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RestrictedCashAndCashEquivalentsCashAndCashEquivalentsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Cash and Cash Equivalents [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherCommitmentsTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Other Commitments [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherCommitmentsTable_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Other Commitments [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationConsiderationTransferred1_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Consideration Transferred</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationConsiderationTransferred1_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total consideration, net of cash acquired</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationConsiderationTransferred1_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Business Combination Consideration Transferred</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RedeemableNoncontrollingInterestEquityPreferredCarryingAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Redeemable Noncontrolling Interest, Equity, Preferred, Carrying Amount</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RedeemableNoncontrollingInterestEquityPreferredCarryingAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Redeemable convertible preferred stock, $0.0001 par value; 100,000 and 51,033 shares authorized as of December 31, 2025 and March 31, 2026, respectively; 0 and 41,588 shares issued and outstanding as of March 31, 2026 and December 31, 2025, respectively; $0 and $1,227,158 liquidation preference as of March 31, 2026 and December 31, 2025, respectively</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NonCashInterestIncome_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Non Cash Interest Income</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NonCashInterestIncome_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Non Cash Interest Income</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NonCashInterestIncome_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Non-cash interest income</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockValue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Preferred Stock, Value, Issued</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockValue_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Preferred stock value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockValue_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Redeemable convertible preferred shares</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionEquityInterestIssuedOrIssuableValueAssigned_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Consideration Transferred, Equity Interest, Share Issued, Value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionEquityInterestIssuedOrIssuableValueAssigned_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Business acquisition in common stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionEquityInterestIssuedOrIssuableValueAssigned_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Equity (issuance of common stock)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ChangeInFairValueOfWarrantLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Change in fair value of warrant liability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ChangeInFairValueOfWarrantLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Change in fair value of warrant liability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ChangeInFairValueOfWarrantLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Change in fair value of warrant liability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ChangeInFairValueOfWarrantLiability_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Change in fair value of warrant liability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SellingGeneralAndAdministrativeExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Selling, General and Administrative Expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SellingGeneralAndAdministrativeExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Selling, General and Administrative Expense, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SellingGeneralAndAdministrativeExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Selling, general and administrative</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SellingGeneralAndAdministrativeExpense_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Selling, general and administrative</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SellingGeneralAndAdministrativeExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Selling, general, and administrative</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LineOfCreditFacilityFairValueOfAmountOutstanding_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Line of Credit Facility, Fair Value of Amount Outstanding</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LineOfCreditFacilityFairValueOfAmountOutstanding_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Amount drawn under the revolving credit facility</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_MinimumMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Minimum [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_MinimumMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Minimum</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsGrantsInPeriodWeightedAverageGrantDateFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Equity Instruments Other than Options, Grants in Period, Weighted Average Grant Date Fair Value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsGrantsInPeriodWeightedAverageGrantDateFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Weighted Average Grant Date Fair Value, Granted</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FinancialInstrumentAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Financial Instrument [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AssetsCurrentAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Assets, Current [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AssetsCurrentAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Current assets:</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TermBLoansMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Term B Loans [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TermBLoansMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Term B Loans [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TermBLoansMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Term B Loans</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AmortizationOfIntangibleAssets_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Amortization of Intangible Assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AmortizationOfIntangibleAssets_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Amortization of acquired intangibles</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonClassBMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Common Class B [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonClassBMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Common Class B</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueInputsLevel3Member_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Value, Inputs, Level 3 [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueInputsLevel3Member_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Level 3</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsFutureAmortizationExpenseAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finite-Lived Intangible Assets, Net, Amortization Expense, Fiscal Year Maturity [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SegmentReportingAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Segment Reporting [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssets_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Recognized Asset Acquired, Asset</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssets_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfFairValueAssetsAndLiabilitiesMeasuredOnRecurringBasisTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule of Fair Value, Assets and Liabilities Measured on Recurring Basis [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockDividendsPerShareDeclared_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Preferred Stock, Dividends Per Share, Declared</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockDividendsPerShareDeclared_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Preferred Stock Dividends Per Share Declared</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LiabilitiesCurrentAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Liabilities, Current [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LiabilitiesCurrentAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Current liabilities:</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LiabilitiesCurrentAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Current:</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Weighted-AverageRemainingContractualTermUnvested_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Weighted-Average Remaining Contractual Term Unvested</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Weighted-AverageRemainingContractualTermUnvested_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Weighted-Average Remaining Contractual Term Unvested</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesJPreferredStockWarrantsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Series J Preferred Stock Warrants</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesJPreferredStockWarrantsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Series J Preferred Stock Warrants [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesJPreferredStockWarrantsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Series J Preferred Stock Warrants [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesJPreferredStockWarrantsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Warrants for Series J Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesSeed-1PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Series Seed-1 Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesSeed-1PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Series Seed-1 Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesSeed-1PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Series Seed-1 Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesD-2PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Series D-2 Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesD-2PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Series D-2 Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesD-2PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Series D-2 Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RelatedPartyTransactionsAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Related Party Transactions [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccountsReceivableNetCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Accounts Receivable, after Allowance for Credit Loss, Current</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccountsReceivableNetCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Accounts receivable, net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccountsReceivableNetCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Accounts receivable, net- related parties</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PrepaidExpenseNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Advanced payments, less current portion</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RestrictedCashAndCashEquivalents_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Total cash and cash equivalents and restricted cash at the end of the period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RestrictedCashEquivalentsCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Restricted Cash Equivalent, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RestrictedCashEquivalentsCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Restricted cash, non-current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommitmentsAndContingencies_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Commitments and Contingencies</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommitmentsAndContingencies_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Commitments and contingencies (See Note 12)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationSeparatelyRecognizedTransactionsAdditionalDisclosuresAcquisitionCosts_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Separately Recognized Transaction, Acquisition-Related Cost, Incurred</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationSeparatelyRecognizedTransactionsAdditionalDisclosuresAcquisitionCosts_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Recognized Cost</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsNonvestedWeightedAverageGrantDateFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Equity Instruments Other than Options, Nonvested, Weighted Average Grant Date Fair Value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsNonvestedWeightedAverageGrantDateFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel">Weighted Average Grant Date Fair Value, Beginning Balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsNonvestedWeightedAverageGrantDateFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel">Weighted Average Grant Date Fair Value, Ending Balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsNonvestedWeightedAverageGrantDateFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Weighted-Average Exercise Price Unvested</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssuedDuringPeriodSharesShareBasedCompensation_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Shares Issued, Shares, Share-Based Payment Arrangement, after Forfeiture</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssuedDuringPeriodSharesShareBasedCompensation_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Shares Issued, Shares, Share-Based Payment Arrangement, after Forfeiture, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssuedDuringPeriodSharesShareBasedCompensation_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Issuance of common stock under equity plan (in shares)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PaymentsToAcquireTimeDeposits_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Payments To Acquire Time Deposits</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PaymentsToAcquireTimeDeposits_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Payments To Acquire Time Deposits</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PaymentsToAcquireTimeDeposits_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Purchases of time deposits</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GeneralAndAdministrativeExpenseMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Selling, general and administrative</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GeneralAndAdministrativeExpenseMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">General and Administrative Expense [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FairMarketValueOfSharePercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Market Value Of Share Percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FairMarketValueOfSharePercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Fair Market Value Of Share Percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FairMarketValueOfSharePercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Fair market value of share, Percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingLeaseLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Operating Lease, Liability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingLeaseLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Operating Lease, Liability, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingLeaseLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Lease Liability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SaleOfStockPercentageOfOwnershipAfterTransaction_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Sale of Stock, Percentage of Ownership after Transaction</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SaleOfStockPercentageOfOwnershipAfterTransaction_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Percentage of shares acquired</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentBasisSpreadOnVariableRate1_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Debt Instrument, Basis Spread on Variable Rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentBasisSpreadOnVariableRate1_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Variable rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockholdersEquityReverseStockSplit_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Stockholders' Equity, Reverse Stock Split</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockholdersEquityReverseStockSplit_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Common stock reverse split</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfGoodwillTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Changes in the Carrying Amount of Goodwill</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfGoodwillTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule of Goodwill [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RestrictedCashCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Restricted Cash, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RestrictedCashCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Restricted cash, current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ConcentrationRiskByBenchmarkAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Concentration Risk Benchmark [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TestStandsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Test stands [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TestStandsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Test stands Member</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TestStandsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Test stands [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DerivativePremiumPaid_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Initial insurance premium</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DerivativePremiumPaid_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Derivative, Premium Paid</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_LeasesTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Leases Text Block</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_LeasesTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Leases Text Block</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_LeasesTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Leases</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ConvertibleNotesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Convertible Notes</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ConvertibleNotesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Convertible Notes [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ConvertibleNotesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Convertible Notes [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInYearTwo_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Long-Term Debt, Maturity, Year Two</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInYearTwo_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">2027</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Share-BasedCompensationArrangementByShare-BasedPaymentAwardOptionsAvailableForGrantGrantsInPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Options Available for Grant, Grants in Period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Share-BasedCompensationArrangementByShare-BasedPaymentAwardOptionsAvailableForGrantGrantsInPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Share-Based Compensation Arrangement by Share-Based Payment Award, Options Available for Grant, Grants in Period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Share-BasedCompensationArrangementByShare-BasedPaymentAwardOptionsAvailableForGrantGrantsInPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Options Available for Grant, Options granted</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FinanceLeaseLiabilityCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Finance lease liability, current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FinanceLeaseLiabilityCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Finance lease liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FinanceLeaseLiabilityCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finance Lease, Liability, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LesseeLeaseDescriptionTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Lessee, Lease, Description [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_WeightedAverageFairValuePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Weighted Average Fair Value Per share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_WeightedAverageFairValuePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Weighted Average Fair Value Per share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_WeightedAverageFairValuePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Weighted Value per Common Warrant</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PercentageOfNonCompoundingRate_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Percentage Of Non Compounding Rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PercentageOfNonCompoundingRate_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Percentage Of Non Compounding Rate.</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PercentageOfNonCompoundingRate_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Percentage Of Non Compounding Rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_LongTermDebtNet_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Long-term debt, net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_LongTermDebtNet_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Long Term Debt Net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_LongTermDebtNet_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Long Term Debt Net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LiabilitiesNoncurrentAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Liabilities, Noncurrent [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LiabilitiesNoncurrentAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Non-current:</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BankTimeDepositsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Bank Time Deposits [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BankTimeDepositsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Time deposits</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CapitalExpendituresIncurredButNotYetPaid_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Capital Expenditures Incurred but Not yet Paid</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CapitalExpendituresIncurredButNotYetPaid_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Non-cash additions of property, plant and equipment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CapitalExpendituresIncurredButNotYetPaid_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Property and equipment additions in accounts payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProfitLoss_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Net Income (Loss), Including Portion Attributable to Noncontrolling Interest</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProfitLoss_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Net loss</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProfitLoss_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Net loss</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCashAndEquivalents_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Recognized Asset Acquired, Cash and Cash Equivalent</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCashAndEquivalents_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Cash and cash equivalents</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCashAndEquivalents_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Cash, net of cash acquired</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SharesOutstandingPercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Shares Outstanding Percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SharesOutstandingPercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Shares Outstanding Percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SharesOutstandingPercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Shares outstanding percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongtermDebtTypeDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Long-Term Debt, Type [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongtermDebtTypeDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Long-Term Debt, Type [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OrganizationalAndOfferingCostsExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Offering and Organizational Costs, Expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OrganizationalAndOfferingCostsExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Offering and Organizational Costs, Expense, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OrganizationalAndOfferingCostsExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Offering costs</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_WeightedAverageNumberOfSharesOutstandingBasic_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Weighted Average Number of Shares Outstanding, Basic</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_WeightedAverageNumberOfSharesOutstandingBasic_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Weighted average common shares outstanding - basic</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_WeightedAverageNumberOfSharesOutstandingBasic_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Basic</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GoodwillAcquiredDuringPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Goodwill, Acquired During Period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GoodwillAcquiredDuringPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Acquisition through business combinations</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccruedPayrollTaxesCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Accrued Payroll Taxes, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccruedPayrollTaxesCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Payroll-related expenses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_StatementScenarioAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Scenario [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_DocumentInformationLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Document Information [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_DocumentInformationLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Document Information [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_TemporaryEquityAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Temporary Equity [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_TemporaryEquityAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Temporary equity</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LineOfCreditFacilityRevolvingCreditConversionToTermLoanDescription_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Line of Credit Facility, Revolving Credit Conversion to Term Loan, Description</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LineOfCreditFacilityRevolvingCreditConversionToTermLoanDescription_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Term loan description</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfShareBasedCompensationArrangementsByShareBasedPaymentAwardTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule of Share-Based Compensation Arrangements by Share-Based Payment Award [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PropertyPlantAndEquipmentTypeDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Long-Lived Tangible Asset [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PropertyPlantAndEquipmentTypeDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Long-Lived Tangible Asset [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNetWorkingCapitalAdjustment_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination Recognized Identifiable Assets Acquired And Liabilities Assumed Net Working Capital Adjustment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNetWorkingCapitalAdjustment_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Business Combination Recognized Identifiable Assets Acquired And Liabilities Assumed Net Working Capital Adjustment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNetWorkingCapitalAdjustment_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Net working capital adjustment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_InterestPayableCurrentAndNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Interest Payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_InterestPayableCurrentAndNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Interest expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SaleAndLeasebackTransactionsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Sale and Leaseback Transactions [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SaleAndLeasebackTransactionsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Sale and Leaseback Transactions [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SaleAndLeasebackTransactionsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Sale and Leaseback Transactions Member</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SubsequentEventsTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Subsequent Events [Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SubsequentEventsTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Subsequent Events</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RelatedPartyTransactionsDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Related Party Transactions Disclosure [Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RelatedPartyTransactionsDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Related Party Transactions</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_VariableRateDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Variable Rate [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_VariableRateDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Variable Rate [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationTotalConsiderationPaid_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total consideration</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationTotalConsiderationPaid_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination Total Consideration Paid</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationTotalConsiderationPaid_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Business Combination Total Consideration Paid</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationTotalConsiderationPaid_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Total consideration</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionProFormaEarningsPerShareDiluted_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Pro forma net loss per common share, Diluted</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionProFormaEarningsPerShareDiluted_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Pro Forma Information, Pro Forma Earnings Per Share, Diluted</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockParOrStatedValuePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Preferred Stock, Par or Stated Value Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockParOrStatedValuePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Preferred Stock, Par or Stated Value Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_InterestIncomeAndInterestExpenseDisclosureTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Interest Income and Interest Expense Disclosure [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_InterestIncomeAndInterestExpenseDisclosureTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Interest Expense Related to the Term Loans</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BasisOfAccountingPolicyPolicyTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Basis of Accounting, Policy [Policy Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BasisOfAccountingPolicyPolicyTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Basis of Presentation</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StatementOfStockholdersEquityAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Statement of Stockholders' Equity [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SeriesCPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Series C Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SeriesCPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Convertible Series C Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SeriesCPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Series C Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementOptionOutstandingFairValueMajoritySponsor_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share Based Compensation Arrangement Option Outstanding Fair Value Majority Sponsor</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementOptionOutstandingFairValueMajoritySponsor_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Share Based Compensation Arrangement Option Outstanding Fair Value Majority Sponsor</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementOptionOutstandingFairValueMajoritySponsor_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Value of Majority Sponsor Top-Up</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseRemainderOfFiscalYear_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finite-Lived Intangible Asset, Expected Amortization, Remainder of Fiscal Year</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseRemainderOfFiscalYear_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">2026 (for the remaining period)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_WarrantsToPurchaseSeriesJPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Warrants to Purchase Series J Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_WarrantsToPurchaseSeriesJPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Warrants to Purchase Series J Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_WarrantsToPurchaseSeriesJPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Warrants to Purchase Series J Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ValuationTechniqueDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Valuation Approach and Technique [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SeriesAPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Series A Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SeriesAPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Series A Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SeriesAPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Convertible Series A Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_UseOfEstimates_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Use of Estimates, Policy [Policy Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_UseOfEstimates_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Use of Estimates</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfDebtInExchangeOfSoftwareLicenses_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Issuance of Debt in Exchange of Software Licenses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfDebtInExchangeOfSoftwareLicenses_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Issuance of Debt in Exchange of Software Licenses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfDebtInExchangeOfSoftwareLicenses_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Issuance of debt in exchange of software licenses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedPropertyPlantAndEquipment_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Recognized Asset Acquired, Property, Plant, and Equipment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedPropertyPlantAndEquipment_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Business Combination, Recognized Asset Acquired, Property, Plant, and Equipment, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedPropertyPlantAndEquipment_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Property and equipment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInYearFive_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Long-Term Debt, Maturity, Year Five</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInYearFive_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">2030</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LeaseholdImprovementsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Leasehold Improvements [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LeaseholdImprovementsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Leasehold improvements [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityAddressAddressLine2_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Entity Address, Address Line Two</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityAddressAddressLine2_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Entity Address, Address Line Two</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueDisclosuresTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Value Disclosures [Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueDisclosuresTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Fair Value Measurement</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationConsiderationTransferredAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Consideration Transferred [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationConsiderationTransferredAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Consideration paid</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfUnderwritingDiscountsAndCommissionsAndOtherOfferingCostsValueDuringPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Issuance Of Underwriting Discounts And Commissions And Other Offering Costs Value During Period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfUnderwritingDiscountsAndCommissionsAndOtherOfferingCostsValueDuringPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Issuance Of Underwriting Discounts And Commissions And Other Offering Costs Value During Period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfUnderwritingDiscountsAndCommissionsAndOtherOfferingCostsValueDuringPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Issuance Of Underwriting Discounts And Commissions And Other Offering Costs</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AccumulatedDeficitMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Accumulated Deficit [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AccumulatedDeficitMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Accumulated Deficit [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Internal-UseSoftwareAndInternally-DevelopedSoftwareMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Internal-use software and Internally-developed software [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Internal-UseSoftwareAndInternally-DevelopedSoftwareMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Internal-use software and Internally-developed software [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Internal-UseSoftwareAndInternally-DevelopedSoftwareMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Internal-use software licenses and software-in-development</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CommonStockSharesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Common Stock Shares [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CommonStockSharesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Common Stock Shares [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentNameDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Debt Instrument, Name [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentNameDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Debt Instrument, Name [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherAssetsCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Other Assets, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherAssetsCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Other current assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NoncashInvestingAndFinancingItemsAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Noncash Investing and Financing Items [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NoncashInvestingAndFinancingItemsAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Supplemental non-cash investing and financing activities:</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NoncashInvestingAndFinancingItemsAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Non-cash investing and financing activities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInYearFour_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Long-Term Debt, Maturity, Year Four</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInYearFour_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">2029</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PreferredStockConvertedToCommonSharesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Preferred Stock Converted To Common Shares</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PreferredStockConvertedToCommonSharesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Preferred Stock Converted To Common Shares [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PreferredStockConvertedToCommonSharesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Preferred stock (as converted to common shares)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SegmentReportingOtherItemAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Segment Reporting, Other Segment Item, Amount</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SegmentReportingOtherItemAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Other segment items</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SegmentReportingOtherItemAmount_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Other segment items</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PlanNameDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Plan Name [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInOtherOperatingAssets_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Increase (Decrease) in Other Operating Assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInOtherOperatingAssets_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel">Other assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInOtherOperatingAssets_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Other assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Value, Assets and Liabilities Measured on Recurring and Nonrecurring Basis [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BuildingMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Buildings [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BuildingMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Building [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CreditFacilityAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Credit Facility [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CreditFacilityAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Credit Facility [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EquityMethodInvestmentOwnershipPercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Equity Method Investment, Ownership Percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EquityMethodInvestmentOwnershipPercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Outstanding common stock owning percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CostOfGoodsAndServicesSold_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Cost of Product and Service Sold</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CostOfGoodsAndServicesSold_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Cost of sales</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CostOfGoodsAndServicesSold_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Cost of Product and Service Sold, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CostOfGoodsAndServicesSold_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Cost of sales</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentRepaidPrincipal_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Outstanding principal amount</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentRepaidPrincipal_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Debt Instrument, Repaid, Principal</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessDescriptionAndBasisOfPresentationTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Description and Basis of Presentation [Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessDescriptionAndBasisOfPresentationTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">The Company, Liquidity, and Basis of Presentation</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CustomerMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Customer [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CustomerMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Customer [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CustomerMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Customer [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FinancingAgreementMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Financing Agreement [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FinancingAgreementMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Financing Agreement [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FinancingAgreementMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Financing Agreement</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_GSPrecisionMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">G.S. Precision [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_GSPrecisionMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">G.S. Precision [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_GSPrecisionMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">G.S. Precision</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AEIndustrialMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">AE Industrial [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AEIndustrialMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">AE Industrial [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AEIndustrialMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">AE Industrial</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FairValuePerShareMarketableSecurity_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Value Per Share Marketable Security</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FairValuePerShareMarketableSecurity_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Fair Value Per Share Marketable Security</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FairValuePerShareMarketableSecurity_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Value per Common Warrant on a marketable basis</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RelatedPartyTransactionsByRelatedPartyAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Related and Nonrelated Parties [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RelatedPartyTransactionsByRelatedPartyAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Related and Nonrelated Parties [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AdditionalPaids-InCapitalNetMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Additional Paids-In Capital Net [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AdditionalPaids-InCapitalNetMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Additional Paids-In Capital Net [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BacklogMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Backlog [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BacklogMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Backlog</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BacklogMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Backlog</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityAddressCityOrTown_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Entity Address, City or Town</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityAddressCityOrTown_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Entity Address, City or Town</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherCommitmentsAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Other Commitments [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherCommitmentsAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Other Commitments [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationContingentConsiderationLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Contingent Consideration, Liability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationContingentConsiderationLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Business Combination, Contingent Consideration, Liability, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationContingentConsiderationLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Contingent liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_SecurityExchangeName_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Security Exchange Name</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_SecurityExchangeName_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Security Exchange Name</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CreditFacilityDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Credit Facility [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CreditFacilityDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Credit Facility [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentCarryingAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Long-Term Debt, Gross</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentCarryingAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Principal</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentCarryingAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Principal amount outstanding</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentCarryingAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PropertyPlantAndEquipmentGross_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Property, Plant and Equipment, Gross</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PropertyPlantAndEquipmentGross_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Property and equipment, gross</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementMaximumValueRecognized_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share Based Compensation Arrangement Maximum Value Recognized</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementMaximumValueRecognized_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Share Based Compensation Arrangement Maximum Value Recognized</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementMaximumValueRecognized_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Amount subject to the Majority Sponsor Top-Up</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IntangibleAssetsDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Intangible Assets, Net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IntangibleAssetsDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Intangible Assets Disclosure [Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Income (Loss) from Continuing Operations before Income Taxes, Noncontrolling Interest</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Loss before provision for income taxes</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Loss before provision for income taxes</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CommonStockOutstandingCommonWarrantsExercised_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Common stock, Outstanding Common Warrants Exercised</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CommonStockOutstandingCommonWarrantsExercised_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Common stock, Outstanding Common Warrants Exercised</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CommonStockOutstandingCommonWarrantsExercised_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Common stock, outstanding common warrants exercised</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ConcentrationRiskTypeDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Concentration Risk Type [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CompensationRelatedCostsAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Compensation Related Costs [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AntidilutiveSecuritiesNameDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Antidilutive Securities, Name [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AntidilutiveSecuritiesNameDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Antidilutive Securities, Name [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Internally-DevelopedSoftwareMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Internally-developed software</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Internally-DevelopedSoftwareMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Internally-developed software [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Internally-DevelopedSoftwareMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Internally-developed software [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockholdersEquityNoteAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Stockholders' Equity Note [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentIncreaseAccruedInterest_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Debt Instrument, Increase, Accrued Interest</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentIncreaseAccruedInterest_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accrued interest</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Cash, Cash Equivalent, Restricted Cash, and Restricted Cash Equivalent, Continuing Operation</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Cash, Cash Equivalent, Restricted Cash, and Restricted Cash Equivalent, Continuing Operation, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Restricted cash, less current portion</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_lbl" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel">Balance, end of period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_lbl" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel">Balance, beginning of period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_InterestPaidNet_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Interest Paid, Excluding Capitalized Interest, Operating Activity</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_InterestPaidNet_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Cash paid for interest</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_RangeMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Statistical Measurement [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_RangeMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Statistical Measurement [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityTransfersNet_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Value, Measurement with Unobservable Inputs Reconciliation, Recurring Basis, Liability, Transfers, Net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityTransfersNet_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Fair Value, Measurement with Unobservable Inputs Reconciliation, Recurring Basis, Liability, Transfers, Net, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityTransfersNet_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Transfers between levels within the fair value hierarchy</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CommonWarrantsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Common Warrants [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CommonWarrantsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Common Warrants [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CommonWarrantsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Common Warrants</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ClassOfWarrantOrRightExercisePriceOfWarrantsOrRights1_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Class of Warrant or Right, Exercise Price of Warrants or Rights</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ClassOfWarrantOrRightExercisePriceOfWarrantsOrRights1_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Exercise price per share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AssetsCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Assets, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AssetsCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total current assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AllocatedShareBasedCompensationExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Payment Arrangement, Expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AllocatedShareBasedCompensationExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Total share-based compensation expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NoncashOrPartNoncashAcquisitionFixedAssetsAcquired1_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Noncash or Part Noncash Acquisition, Fixed Assets Acquired</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NoncashOrPartNoncashAcquisitionFixedAssetsAcquired1_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Property and equipment acquired through loans</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_DocumentPeriodEndDate_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Document Period End Date</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_DocumentPeriodEndDate_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Document Period End Date</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueMeasurementsFairValueHierarchyDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Value Hierarchy and NAV [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationContingentConsiderationAsset_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Contingent Consideration, Asset</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationContingentConsiderationAsset_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Business Combination, Contingent Consideration, Asset, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationContingentConsiderationAsset_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Business acquisition total consideration</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CommonStockWarrantMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Common Stock Warrant [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CommonStockWarrantMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Common Stock Warrant [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CommonStockWarrantMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Common Warrant</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Exercisable, Number</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Number of Options Outstanding, Vested and exercisable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LineOfCreditFacilityCommitmentFeePercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Line of Credit Facility, Commitment Fee Percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LineOfCreditFacilityCommitmentFeePercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Commitment fee</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AcquiredFiniteLivedIntangibleAssetsWeightedAverageUsefulLife_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Acquired Finite-Lived Intangible Assets, Weighted Average Useful Life</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AcquiredFiniteLivedIntangibleAssetsWeightedAverageUsefulLife_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Weighted Average Life</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StatementOfFinancialPositionAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Statement of Financial Position [Abstract]</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RestrictedStockUnitsRSUMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Restricted Stock Units (RSUs) [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RedeemableConvertiblePreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Redeemable Convertible Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RedeemableConvertiblePreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Redeemable Convertible Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ScheduleOfRedeemableConvertiblePreferredStock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule Of Redeemable Convertible Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ScheduleOfRedeemableConvertiblePreferredStock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule Of Redeemable Convertible Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ScheduleOfRedeemableConvertiblePreferredStock_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Schedule Of Redeemable Convertible Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Antidilutive Security, Excluded EPS Calculation [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTable_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Antidilutive Security, Excluded EPS Calculation [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_VestingAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Vesting [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StatementClassOfStockAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Class of Stock [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StatementClassOfStockAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Class of Stock [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FairValueInputsDiscountRate_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Discount factor</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FairValueInputsDiscountRate_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Value Inputs Discount Rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FairValueInputsDiscountRate_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Fair Value Inputs Discount Factor</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_RedeemableConvertiblePreferredStocksMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Redeemable Convertible Preferred Stocks [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_RedeemableConvertiblePreferredStocksMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Redeemable Convertible Preferred Stocks</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_RedeemableConvertiblePreferredStocksMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Redeemable Convertible Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NonCancelableServiceContractCommitmentsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Non-Cancelable Service Contract Commitments</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NonCancelableServiceContractCommitmentsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Non-Cancelable Service Contract Commitments [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NonCancelableServiceContractCommitmentsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Non-Cancelable Service Contract Commitments</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_RevisionOfPriorPeriodReclassificationAdjustmentMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Revision of Prior Period, Reclassification, Adjustment [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_RevisionOfPriorPeriodReclassificationAdjustmentMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Adjustments</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_Liabilities_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_Liabilities_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_TransfersAndServicingOfFinancialInstrumentsTypesOfFinancialInstrumentsDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Financial Instruments [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FourPointFiveZeroPercentDueTwoThousandTwentySixMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Four Point Five Zero Percent Due Two Thousand Twenty six [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FourPointFiveZeroPercentDueTwoThousandTwentySixMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Four Point Five Zero Percent Due Two Thousand Twenty six [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FourPointFiveZeroPercentDueTwoThousandTwentySixMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">4.50% due 2026</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PaymentsForTaxesRelatedToNetShareSettlementOfEquityAwards_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Payments For Taxes Related To Net Share Settlement Of Equity Awards</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PaymentsForTaxesRelatedToNetShareSettlementOfEquityAwards_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Payments for taxes related to net share settlement of equity awards</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PaymentsForTaxesRelatedToNetShareSettlementOfEquityAwards_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Payments for taxes related to net share settlement of equity awards.</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ValuationTechniqueOptionPricingModelMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">forward pricing valuation model</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ValuationTechniqueOptionPricingModelMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Valuation Technique, Option Pricing Model [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInDeferredRevenue_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Deferred revenue</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInDeferredRevenue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Increase (Decrease) in Deferred Revenue</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LossContingencyEstimateOfPossibleLoss_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Loss Contingency, Estimate of Possible Loss</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LossContingencyEstimateOfPossibleLoss_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Loss contingency, estimate of possible loss</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtConversionConvertedInstrumentWarrantsOrOptionsIssued1_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Debt Conversion, Converted Instrument, Warrants or Options Issued</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtConversionConvertedInstrumentWarrantsOrOptionsIssued1_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Issued to the lenders series J warrants to purchase</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_OneTimeCostsRelatedToTheIPO_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">One-time costs related to the IPO</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_OneTimeCostsRelatedToTheIPO_lbl" xlink:role="http://www.xbrl.org/2003/role/label">One Time Costs Related To The IPO</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_OneTimeCostsRelatedToTheIPO_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">One Time Costs Related To The IPO</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RestrictedCash_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Restricted Cash</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RestrictedCash_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Restricted Cash, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RestrictedCash_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Restricted cash, current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RestrictedCash_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Restricted cash balances</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RestrictedStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Restricted Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredRevenueArrangementByTypeTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Deferred Revenue Arrangement, by Type [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FinanceLeaseRightOfUseAssetAmortization_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finance Lease, Right-of-Use Asset, Amortization</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FinanceLeaseRightOfUseAssetAmortization_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Amortization of ROU assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FinanceLeaseRightOfUseAssetAmortization_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Amortization of right-of-use assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_VariableRateAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Variable Rate [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_VariableRateAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Variable Rate [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IndefiniteLivedIntangibleAssetsMajorClassNameDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Indefinite-Lived Intangible Assets, Major Class Name [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsGross_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Gross Carrying Amount</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsGross_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finite-Lived Intangible Assets, Gross</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsGross_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Finite-Lived Intangible Assets, Gross, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Weighted Average Number of Shares Outstanding, Diluted</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Weighted average common shares outstanding - diluted</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Diluted</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityFileNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Entity File Number</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityFileNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Entity File Number</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CustomerThreeMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Customer Three [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CustomerThreeMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Customer Three Member</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_CustomerThreeMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Customer 3</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NumberOfReservedSharesIncreaseApproximately_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Number Of Reserved Shares, Increase approximately</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NumberOfReservedSharesIncreaseApproximately_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Number Of Reserved Shares, Increase approximately</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NumberOfReservedSharesIncreaseApproximately_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Number Of Reserved Shares, Increase approximately</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_InternalUseSoftwareMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Internal Use Software [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_InternalUseSoftwareMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Internal Use Software</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_InternalUseSoftwareMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Internal-use software licenses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_CoverAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Cover [Abstract]</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfCarryingValuesAndEstimatedFairValuesOfDebtInstrumentsTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule of Carrying Values and Estimated Fair Values of Debt Instruments [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfCarryingValuesAndEstimatedFairValuesOfDebtInstrumentsTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Carrying Value of The Term Loans Approximated Their Estimated Fair Value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsForfeitedInPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Equity Instruments Other than Options, Forfeited in Period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsForfeitedInPeriod_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Number of shares, Canceled / forfeited</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsForfeitedInPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Number of shares, Canceled / forfeited</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CapitalLeaseObligationsIncurred_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Lease Obligation Incurred</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CapitalLeaseObligationsIncurred_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Property and equipment acquired through finance leases</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentLiabilitiesOther_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Other current liabilities</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SalesRevenueNetMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Revenue Benchmark [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SalesRevenueNetMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Revenue Benchmark [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_ScenarioUnspecifiedDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Scenario [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_RPMCallOptionMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">RPM Call Option [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_RPMCallOptionMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">RPM Call Option [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_RPMCallOptionMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">RPM Call Option</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NumberOfOperatingSegments_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Number of Operating Segments</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SaleOfStockNameOfTransactionDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Sale of Stock [Domain]</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_EquityMethodInvestmentMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Equity Method Investment [Member]</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_LaunchRevenueMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Launch Revenue [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_LaunchRevenueMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Launch Revenue [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_LaunchRevenueMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Launch Revenue</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SegmentReportingDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Segment Reporting Disclosure [Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SegmentReportingDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Segment and Geographical Information</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfAcquiredFiniteLivedIntangibleAssetByMajorClassTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule of Acquired Finite-Lived Intangible Asset by Major Class [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SeriesBPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Series B Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SeriesBPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Series B Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SeriesBPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Convertible Series B Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NonvestedRestrictedStockSharesActivityTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Nonvested Restricted Stock Shares Activity [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NonvestedRestrictedStockSharesActivityTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Summary of restricted stock activity</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RevenueFromContractWithCustomerIncludingAssessedTax_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Revenue from Contract with Customer, Including Assessed Tax</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RevenueFromContractWithCustomerIncludingAssessedTax_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Revenue</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_TemporaryEquityLiquidationPreference_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Temporary Equity, Liquidation Preference</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_TemporaryEquityLiquidationPreference_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Aggregate Liquidation Preferences</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractualObligationNumberOfLaunches_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Contractual Obligation, Number Of Launches</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractualObligationNumberOfLaunches_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Contractual Obligation, Number Of Launches</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractualObligationNumberOfLaunches_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Loss contingency, number of launch</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SubsequentEventTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Subsequent Event [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GeographicAreasRevenuesFromExternalCustomersAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Geographic Areas, Revenues from External Customers [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionsProFormaNetIncomeLoss_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Business combination post-acquisition income and loss</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionsProFormaNetIncomeLoss_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Pro Forma Information, Pro Forma Income (Loss), after Tax</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_OperatingLeaseRight-Of-UseAssetDecreased_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Operating Lease, Right-of-Use Asset Decreased</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_OperatingLeaseRight-Of-UseAssetDecreased_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Operating Lease, Right-of-Use Asset Decreased</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_OperatingLeaseRight-Of-UseAssetDecreased_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Right-of-use assets - operating leases decreases</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredCostsCurrentAndNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Deferred Costs</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredCostsCurrentAndNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Deferred Costs, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredCostsCurrentAndNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Deferred contract costs</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredCostsCurrentAndNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Retainage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentLiabilitiesAccountsPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accounts payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentLiabilitiesAccountsPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Recognized Liability Assumed, Accounts Payable, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsNet_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finite-Lived Intangible Assets, Net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsNet_lbl" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel">Finite-Lived Intangible Assets, Net, Ending Balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsNet_lbl" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel">Finite-Lived Intangible Assets, Net, Beginning Balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsNet_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Net Carrying Amount</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsNet_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Total future amortization</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsNet_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total future amortization</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LineOfCreditFacilityInterestRateDuringPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Line of Credit Facility, Interest Rate During Period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LineOfCreditFacilityInterestRateDuringPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Interest rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AdjustmentsToAdditionalPaidInCapitalWarrantIssued_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Adjustments to Additional Paid in Capital, Warrant Issued</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AdjustmentsToAdditionalPaidInCapitalWarrantIssued_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Purchase of an Aggregate of Common Warrant</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GainsLossesOnExtinguishmentOfDebt_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Loss on extinguishment of debt</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GainsLossesOnExtinguishmentOfDebt_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Gain (Loss) on Extinguishment of Debt</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GainsLossesOnExtinguishmentOfDebt_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Gain (Loss) on Extinguishment of Debt, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GainsLossesOnExtinguishmentOfDebt_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Loss on extinguishment of debt</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GainsLossesOnExtinguishmentOfDebt_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel">Loss on extinguishment of debt</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsNonvestedWeightedAverageGrantDateFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Option, Nonvested, Weighted Average Exercise Price</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsNonvestedWeightedAverageGrantDateFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel">Share-Based Compensation Arrangement by Share-Based Payment Award, Option, Nonvested, Weighted Average Exercise Price, Ending Balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsNonvestedWeightedAverageGrantDateFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel">Share-Based Compensation Arrangement by Share-Based Payment Award, Option, Nonvested, Weighted Average Exercise Price, Beginning Balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NewAccountingPronouncementsPolicyPolicyTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">New Accounting Pronouncements, Policy [Policy Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NewAccountingPronouncementsPolicyPolicyTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Recent Accounting Pronouncements</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Antidilutive Securities Excluded from Computation of Earnings Per Share [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Antidilutive Securities Excluded from Computation of Earnings Per Share [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GainLossOnDispositionOfAssets1_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Gain (Loss) on Disposition of Assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GainLossOnDispositionOfAssets1_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Gain (Loss) on Disposition of Assets, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GainLossOnDispositionOfAssets1_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Loss on sale of fixed assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GainLossOnDispositionOfAssets1_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">(Gain) loss on disposal of fixed assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationCashConsiderationPaid_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination Cash Consideration Paid</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationCashConsiderationPaid_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Business Combination Cash Consideration Paid</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationCashConsiderationPaid_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Cash</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfComprehensiveIncomeLossTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Comprehensive Income (Loss) [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfComprehensiveIncomeLossTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of net loss and comprehensive loss on date of acquisition</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLiabilities_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Recognized Liability Assumed, Liability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLiabilities_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NonrecurringAdjustmentAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Pro Forma Information, Nonrecurring Adjustment [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_Depreciation_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Depreciation</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_Depreciation_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Depreciation expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognizedShareBasedAwardsOtherThanOptions_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Payment Arrangement, Nonvested Award, Excluding Option, Cost Not yet Recognized, Amount</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognizedShareBasedAwardsOtherThanOptions_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Unrecognized compensation expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ROUAssetsExchangeOfLeaseLiabilities_lbl" xlink:role="http://www.xbrl.org/2003/role/label">ROU Assets Exchange of Lease Liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ROUAssetsExchangeOfLeaseLiabilities_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">ROU Assets Exchange of Lease Liabilites</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ROUAssetsExchangeOfLeaseLiabilities_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">ROU assets exchange of lease liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonStockSharesOutstanding_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Common Stock, Shares, Outstanding</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonStockSharesOutstanding_lbl" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel">Common stock, shares, outstanding, beginning balance (in shares)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonStockSharesOutstanding_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Common stock, shares, outstanding (in shares)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingLeaseLiabilityCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Operating Lease, Liability, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingLeaseLiabilityCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Operating lease liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingLeaseLiabilityCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Operating lease liability, current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DepreciationDepletionAndAmortization_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Depreciation, Depletion and Amortization, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DepreciationDepletionAndAmortization_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Depreciation, Depletion and Amortization</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DepreciationDepletionAndAmortization_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Depreciation and amortization</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RevenueRemainingPerformanceObligation_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Revenue, Remaining Performance Obligation, Amount</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RevenueRemainingPerformanceObligation_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Revenue, remaining performance obligation</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_DocumentTransitionReport_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Document Transition Report</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_DocumentTransitionReport_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Document Transition Report</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShortTermInvestments_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Short-Term Investments</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShortTermInvestments_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Short-term investments</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingExpenses_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Operating Expenses</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ResearchAndDevelopmentExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Research and Development Expense</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ProceedsFromNotesPayableRelatedParties_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Proceeds From Notes Payable, Related Parties</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ProceedsFromNotesPayableRelatedParties_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Proceeds from notes payable - related parties</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ConvertiblePreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Convertible Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ConvertiblePreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Convertible Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseRollingYearThree_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finite-Lived Intangible Assets, Amortization Expense, Rolling Year Three</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseRollingYearThree_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">2029</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StatementOfCashFlowsAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Statement of Cash Flows [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MinimumCashBalanceForPurchaseAndLeasebackAgreements_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Minimum Cash Balance for Purchase and Leaseback Agreements</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MinimumCashBalanceForPurchaseAndLeasebackAgreements_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Minimum Cash Balance for Purchase and Leaseback Agreements</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MinimumCashBalanceForPurchaseAndLeasebackAgreements_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Minimum cash balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_OtherNotesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Other notes [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_OtherNotesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Other notes [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_OtherNotesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Other notes</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractWithCustomerLossProvision_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Contract With Customer Loss Provision</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractWithCustomerLossProvision_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Contract With Customer Loss Provision</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractWithCustomerLossProvision_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Contract With Customer Loss Provision</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssuedDuringPeriodValueNewIssues_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Stock Issued During Period, Value, New Issues</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssuedDuringPeriodValueNewIssues_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Issuance of stock, net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfShareBasedCompensationStockOptionsActivityTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Summary of stock option activity</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfShareBasedCompensationStockOptionsActivityTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Payment Arrangement, Option, Activity [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BrokerageCommissionsRevenue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Brokerage Commissions Revenue</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BrokerageCommissionsRevenue_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Amount of commission</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueOptionChangesInFairValueGainLoss1_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Recognized loss</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueOptionChangesInFairValueGainLoss1_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Value, Option, Changes in Fair Value, Gain (Loss)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsToAcquireBusinessesNetOfCashAcquired_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Payments to Acquire Businesses, Net of Cash Acquired</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsToAcquireBusinessesNetOfCashAcquired_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Payments to Acquire Businesses, Net of Cash Acquired, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsToAcquireBusinessesNetOfCashAcquired_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Acquisition of business, net of acquired cash</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockholdersEquityNoteDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Equity [Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockholdersEquityNoteDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Stockholders' Equity and Redeemable Convertible Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PaymentForPurchaseAndLeasebackAgreements_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Payment for Purchase and Leaseback Agreements</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PaymentForPurchaseAndLeasebackAgreements_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Payment for Purchase and Leaseback Agreements</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PaymentForPurchaseAndLeasebackAgreements_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Payment for purchase and leaseback agreements</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SpaceflightMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Spaceflight [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SpaceflightMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Spaceflight [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SpaceflightMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Spaceflight</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ThirteenPointEightHundredSeventyFivePercentDueTwoThousandTwentyEightMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Thirteen Point Eight Hundred Seventy Five Percent Due Two Thousand Twenty Eight [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ThirteenPointEightHundredSeventyFivePercentDueTwoThousandTwentyEightMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Thirteen Point Eight Hundred Seventy Five Percent Due Two Thousand Twenty Eight [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ThirteenPointEightHundredSeventyFivePercentDueTwoThousandTwentyEightMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">13.875% due 2028</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finite-Lived Intangible Assets [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EarningsPerShareTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Earnings Per Share [Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EarningsPerShareTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Net Loss Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SharesIssued_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Shares, Issued</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SharesIssued_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Shares Issued</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationGoodwillAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Goodwill [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfShareBasedCompensationRestrictedStockUnitsAwardActivityTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Payment Arrangement, Restricted Stock Unit, Activity [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfShareBasedCompensationRestrictedStockUnitsAwardActivityTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Summary of the RSU activity</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ComputerEquipmentMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Computer Equipment [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ComputerEquipmentMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Computer Equipment [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetIncomeLoss_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Consolidated net loss</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetIncomeLoss_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Net loss and comprehensive loss</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingIntrinsicValue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Outstanding, Intrinsic Value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingIntrinsicValue_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Aggregate intrinsic value of the options outstanding</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RealEstateInventoryCapitalizedInterestCosts_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Real Estate Inventory, Capitalized Interest Costs</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RealEstateInventoryCapitalizedInterestCosts_lbl" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel">Real Estate Inventory, Capitalized Interest Costs, Ending Balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RealEstateInventoryCapitalizedInterestCosts_lbl" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel">Real Estate Inventory, Capitalized Interest Costs, Beginning Balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RealEstateInventoryCapitalizedInterestCosts_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Capitalized interest in connection with construction in progress</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DividendsPreferredStockCash_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Dividends, Preferred Stock, Cash</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DividendsPreferredStockCash_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Dividends, Preferred Stock, Cash</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingLeaseCost_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Operating Lease, Cost</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingLeaseCost_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Operating lease expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedDividendRate_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Dividend yield</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedDividendRate_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Fair Value Assumptions, Expected Dividend Rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EarningsPerShareBasicLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Earnings Per Share, Basic, by Common Class, Including Two Class Method [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EarningsPerShareBasicLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Earnings Per Share, Basic, by Common Class, Including Two Class Method [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IntangibleAssetsNetExcludingGoodwill_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Intangible Assets, Net (Excluding Goodwill)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IntangibleAssetsNetExcludingGoodwill_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Intangible assets, net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IntangibleAssetsNetExcludingGoodwill_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Intangible Assets, Net (Excluding Goodwill), Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_OtherAddressMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Other Address [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LineOfCreditFacilityLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Line of Credit Facility [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DiscountsOnUnderwriting_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Discounts on Underwriting</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DiscountsOnUnderwriting_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Discounts on Underwriting</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DiscountsOnUnderwriting_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Discounts on Underwriting</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AssetsAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Assets [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AssetsAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">ASSETS</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AssetsAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Assets:</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DividendsPreferredStockStock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Dividends, Preferred Stock, Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsRiskFreeInterestRate_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Fair Value Assumptions, Risk Free Interest Rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsRiskFreeInterestRate_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Risk-free rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsRiskFreeInterestRate_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Risk-free interest rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ConstructionInProgressMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Construction in Progress [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ConstructionInProgressMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Construction in progress [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NumberOfReservedSharesIncrease_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Number Of Reserved Shares, Increase</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NumberOfReservedSharesIncrease_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Number Of Reserved Shares, Increase</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_NumberOfReservedSharesIncrease_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Increment in reserved shares</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationEquityConsiderationPaid_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination Equity Consideration Paid</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationEquityConsiderationPaid_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Business Combination Equity Consideration Paid</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationEquityConsiderationPaid_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Equity</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_DocumentFiscalYearFocus_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Document Fiscal Year Focus</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_DocumentFiscalYearFocus_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Document Fiscal Year Focus</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtConversionConvertedInstrumentSharesIssued1_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Debt Conversion, Converted Instrument, Shares Issued</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtConversionConvertedInstrumentSharesIssued1_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Preferred Stock settled shares</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ConcentrationRiskBenchmarkDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Concentration Risk Benchmark [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Cash Provided by (Used in) Investing Activity, Including Discontinued Operation [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Cash Flows from investing activities:</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccountsPayableCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Accounts Payable, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccountsPayableCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Accounts payable- related parties</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccountsPayableCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accounts payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SaleLeasebackTransactionDescriptionAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Sale Leaseback Transaction, Description [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_InterestExpenseNonoperating_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Interest Expense, Nonoperating</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_InterestExpenseNonoperating_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel">Finance and interest expense, net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_InterestExpenseNonoperating_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Interest Expense, Nonoperating, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_InterestExpenseNonoperating_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Interest expense, net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_InterestExpenseNonoperating_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Interest expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsOfDividendsPreferredStockAndPreferenceStock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Payments of Ordinary Dividends, Preferred Stock and Preference Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsOfDividendsPreferredStockAndPreferenceStock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Payment of IPO Closing Preferred Stock Dividend</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsOfDividendsPreferredStockAndPreferenceStock_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Closing Preferred Stock Dividend</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsOfDividendsPreferredStockAndPreferenceStock_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel">Closing Preferred Stock Dividend</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsOfDividendsPreferredStockAndPreferenceStock_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Payment of initial public offering closing Preferred Stock Dividend</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ResearchAndDevelopmentExpenseMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Research and development</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ResearchAndDevelopmentExpenseMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Research and Development Expense [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_EventProbabilityPercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Event Probability Percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_EventProbabilityPercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Event Probability Percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_EventProbabilityPercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Event weighting</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ContractWithCustomerAssetAndLiabilityTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Contract with Customer, Contract Asset, Contract Liability, and Receivable [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ContractWithCustomerAssetAndLiabilityTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Contract Assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredRevenue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Deferred Revenue</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredRevenue_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Advanced payments, less current portion</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredRevenue_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Total deferred revenue</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredRevenue_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total deferred revenue</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssuedDuringPeriodValueShareBasedCompensation_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Shares Issued, Value, Share-Based Payment Arrangement, after Forfeiture</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssuedDuringPeriodValueShareBasedCompensation_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Shares Issued, Value, Share-Based Payment Arrangement, after Forfeiture, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssuedDuringPeriodValueShareBasedCompensation_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Issuance of common stock under equity plan</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DerivativeBasisSpreadOnVariableRate_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Variable rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DerivativeBasisSpreadOnVariableRate_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Derivative, Basis Spread on Variable Rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of future amortization expense on intangible assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finite-Lived Intangible Assets Amortization Expense [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_RestatementDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Revision of Prior Period [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesD-1PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Series D-1 Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesD-1PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Series D-1 Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesD-1PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Convertible Series D-1 Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesD-1PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Series D-1 Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromStockOptionsExercised_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Proceeds from Stock Options Exercised</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromStockOptionsExercised_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Share exercised over allotment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromStockOptionsExercised_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Proceeds from exercise of stock options</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueDisclosuresAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Value Disclosures [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_VotingPowerPercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Voting Power</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_VotingPowerPercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Voting Power Percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_VotingPowerPercentage_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Voting Power Percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommitmentsAndContingenciesDisclosureAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Commitments and Contingencies Disclosure [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Non-CashInvestingAndFinancingActivitiesEquityIssuedForBusinessCombination_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Non-cash Investing and Financing Activities Equity Issued for Business Combination</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Non-CashInvestingAndFinancingActivitiesEquityIssuedForBusinessCombination_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Non-cash Investing and Financing Activities Equity Issued for Business Combination</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Non-CashInvestingAndFinancingActivitiesEquityIssuedForBusinessCombination_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Equity issued for business combination</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SaleOfStockConsiderationReceivedOnTransaction_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Sale of Stock, Consideration Received on Transaction</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SaleOfStockConsiderationReceivedOnTransaction_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Aggregate purchase price</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SaleOfStockConsiderationReceivedOnTransaction_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Series D-1 Preferred Stock funding</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Net Income (Loss) Available to Common Stockholders, Basic</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Net loss available to common stockholders, basic</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentLiabilitiesDeferredRevenue_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Deferred revenue</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentLiabilitiesDeferredRevenue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Recognized Liability Assumed, Deferred Revenue, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNet_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Recognized Asset Acquired to Liability Assumed, Excess (Less)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNet_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Net assets acquired</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedGoodwill_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination Recognized Identifiable Assets Acquired And Liabilities Assumed Goodwill</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedGoodwill_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Business Combination Recognized Identifiable Assets Acquired And Liabilities Assumed Goodwill</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedGoodwill_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Goodwill</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LineOfCredit_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Long-Term Line of Credit</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LineOfCredit_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Long-Term Line of Credit, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LineOfCredit_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Revolving credit facility, aggregate principal amounts</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionsProFormaIncomeLossFromContinuingOperationsBeforeChangesInAccountingAndExtraordinaryItemsNetOfTax_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Pro Forma Information, Pro Forma Income (Loss) from Continuing Operations, after Tax</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionsProFormaIncomeLossFromContinuingOperationsBeforeChangesInAccountingAndExtraordinaryItemsNetOfTax_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Pro forma net loss and comprehensive loss</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionProFormaInformationTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Unaudited Pro Forma Information</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionProFormaInformationTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Pro Forma Information [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Debt Instrument [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Debt Instrument [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DerivativeInstrumentRiskAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Derivative Instrument [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DisclosureOfCompensationRelatedCostsSharebasedPaymentsAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Payment Arrangement [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AssetsAndLiabilitiesLesseeTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Assets And Liabilities Lessee [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AssetsAndLiabilitiesLesseeTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Assets And Liabilities Lessee [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AssetsAndLiabilitiesLesseeTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Operating and Finance Lease Right-of-Use Assets and Liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognized_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Payment Arrangement, Nonvested Award, Cost Not yet Recognized, Amount</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognized_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Share-Based Payment Arrangement, Nonvested Award, Cost Not yet Recognized, Amount, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognized_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Unrecognized compensation cost</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AssetsFairValueDisclosure_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Total financial assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AssetsFairValueDisclosure_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Assets, Fair Value Disclosure</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AssetsFairValueDisclosure_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Assets, Fair Value Disclosure, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SubsidiarySaleOfStockAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Sale of Stock [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SubsidiarySaleOfStockAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Sale of Stock [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedDeferredTaxLiabilities_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Deferred tax liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedDeferredTaxLiabilities_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Recognized Liability Assumed, Deferred Tax Liability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ConcentrationRiskPercentage1_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Concentration Risk, Percentage</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DevelopedTechnologyMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Developed Technology [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DevelopedTechnologyMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Developed Technology [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DevelopedTechnologyMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Developed technology</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_Revenues_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Revenue</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfCommonStockToSettlePreferredStockDividendShares_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Issuance of Common Stock to Settle Preferred Stock Dividend Shares</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_OtherArrangementsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Other Arrangements [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_OtherArrangementsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Other Arrangements</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesD-4PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Series D-4 Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesD-4PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Series D-4 Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfSegmentReportingInformationBySegmentTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule of Segment Reporting Information, by Segment [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfSegmentReportingInformationBySegmentTable_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Segment Reporting Information, by Segment [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ExerciseOfCommonWarrantsIntoCommonStock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Net exercise of Common Warrants into common stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ExerciseOfCommonWarrantsIntoCommonStock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Exercise of Common Warrants into Common Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ExerciseOfCommonWarrantsIntoCommonStock_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Exercise of Common Warrants into Common Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsForfeituresWeightedAverageGrantDateFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Equity Instruments Other than Options, Forfeitures, Weighted Average Grant Date Fair Value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsForfeituresWeightedAverageGrantDateFairValue_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Weighted Average Grant Date Fair Value, Canceled / Forfeited</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_MoneyMarketFundsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Money Market Funds [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_MoneyMarketFundsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Money market funds</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RelatedPartyDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Related and Nonrelated Parties [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RelatedPartyDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Related and Nonrelated Parties [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseRollingYearFive_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finite-Lived Intangible Assets, Amortization Expense, Rolling Year Five</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseRollingYearFive_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Thereafter</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalAfterYearFive_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Long-Term Debt, Maturity, after Year Five</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalAfterYearFive_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Thereafter</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncomeTaxDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Income Tax Disclosure [Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncomeTaxDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Income Taxes</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfCollaborativeArrangementsAndNoncollaborativeArrangementTransactionsTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Collaborative Arrangement and Arrangement Other than Collaborative [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_TemporaryEquitySharesIssued_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Temporary Equity, Shares Issued</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_TemporaryEquitySharesIssued_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Temporary equity, Shares issued</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DebtInstrumentCarryingAmountNetOfIssuanceCosts_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Total debt, net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DebtInstrumentCarryingAmountNetOfIssuanceCosts_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Debt Instrument Carrying Amount Net Of Issuance Costs</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DebtInstrumentCarryingAmountNetOfIssuanceCosts_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Debt Instrument Carrying Amount Net Of Issuance Costs</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DebtInstrumentCarryingAmountNetOfIssuanceCosts_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total debt, net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_DocumentInformationTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Document Information [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_DocumentInformationTable_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Document Information [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherLiabilitiesNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Other Liabilities, Noncurrent</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherLiabilitiesNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Other liabilities, less current portion</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FinanceLeaseRightOfUseAsset_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finance Lease, Right-of-Use Asset, after Accumulated Amortization</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FinanceLeaseRightOfUseAsset_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Finance Lease, Right-of-Use Asset, after Accumulated Amortization, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FinanceLeaseRightOfUseAsset_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Right-of-use assets - finance leases</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FinanceLeaseRightOfUseAsset_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Finance lease right-of-use assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInOperatingCapitalAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Adjustment to Reconcile Net Income to Cash Provided by (Used in) Operating Activity, Increase (Decrease) in Operating Capital [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncreaseDecreaseInOperatingCapitalAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Changes in operating assets and liabilities:</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PropertyPlantAndEquipmentTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Property, Plant and Equipment [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PropertyPlantAndEquipmentTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Property and Equipment, Net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LettersOfCreditOutstandingAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Letters of Credit Outstanding, Amount</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LettersOfCreditOutstandingAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Letters of credit</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Outstanding, Weighted Average Exercise Price</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel">Weighted Average Exercise Price , Ending Balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel">Weighted Average Exercise Price, Beginning Balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncomeTaxDisclosureAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Income Tax Disclosure [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PercentageOfNonAccruingDividendDeemedIssuePricePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Percentage Of Non Accruing Dividend Deemed Issue Price Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PercentageOfNonAccruingDividendDeemedIssuePricePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Percentage Of Non Accruing Dividend Deemed Issue Price Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PercentageOfNonAccruingDividendDeemedIssuePricePerShare_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Percentage of Non Accruing Dividend Deemed Issue Price Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Value, Recurring and Nonrecurring [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Business Combinations</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination [Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AdditionalPaidInCapitalMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Additional Paid-in Capital [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AdditionalPaidInCapitalMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Additional Paid-in Capital, Net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Internally-DevelopedSoftwareNotYetSubjectToAmortization_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Internally-developed software not yet subject to amortization</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Internally-DevelopedSoftwareNotYetSubjectToAmortization_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Internally-developed software not yet subject to amortization</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Internally-DevelopedSoftwareNotYetSubjectToAmortization_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Internally-developed software not yet subject to amortization</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FairValuePerShareNonMarketableSecurity_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Value Per Share Non Marketable Security</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FairValuePerShareNonMarketableSecurity_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Fair Value Per Share Non Marketable Security</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FairValuePerShareNonMarketableSecurity_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Value per Common Warrant on a non-marketable basis</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncomeStatementLocationAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Statement of Income Location, Balance [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_TradingSymbol_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Trading Symbol</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_TradingSymbol_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Trading Symbol</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EquityMethodInvestmentsFairValueDisclosure_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Equity Method Investments, Fair Value Disclosure</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EquityMethodInvestmentsFairValueDisclosure_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Fair value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PropertyPlantAndEquipmentDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Property, Plant and Equipment Disclosure [Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PropertyPlantAndEquipmentDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Property and Equipment, Net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LiabilitiesFairValueAdjustment_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Liabilities, Fair Value Adjustment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LiabilitiesFairValueAdjustment_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Fair value warrant liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfRelatedPartyTransactionsByRelatedPartyTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Related Party Transaction [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfRelatedPartyTransactionsByRelatedPartyTable_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Related Party Transaction [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromNotesPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Proceeds from Notes Payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromNotesPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Proceeds from Notes Payable, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromNotesPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Proceeds from issuance of notes payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SubsequentEventTypeAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Subsequent Event Type [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfFiniteLivedIntangibleAssetsTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule of Finite-Lived Intangible Assets [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfFiniteLivedIntangibleAssetsTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of intangible assets gross carrying amount and accumulated amortization</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementOptionFairValueAssumption_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Black-Scholes value (per share)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementOptionFairValueAssumption_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share Based Compensation Arrangement Option Fair Value Assumption</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementOptionFairValueAssumption_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Share Based Compensation Arrangement Option Fair Value Assumption</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Cash Provided by (Used in) Financing Activity, Including Discontinued Operation [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Cash Flows from financing activities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardFairValueAssumptionsExpectedTerm1_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Term (years)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardFairValueAssumptionsExpectedTerm1_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Fair Value Assumptions, Expected Term</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAccruedExpenses_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination Recognized Identifiable Assets Acquired and Liabilities Assumed Accrued Expenses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAccruedExpenses_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Business Combination Recognized Identifiable Assets Acquired and Liabilities Assumed Accrued Expenses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAccruedExpenses_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accrued expenses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccountsPayableAndAccruedLiabilitiesDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Accounts Payable and Accrued Liabilities Disclosure [Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccountsPayableAndAccruedLiabilitiesDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accrued Expenses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityExTransitionPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Entity Ex Transition Period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityExTransitionPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Entity Ex Transition Period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfPreferredStockDividendValue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Issuance of Preferred Stock Dividend Value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfPreferredStockDividendValue_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Issuance of Preferred Stock Dividend Value</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfPreferredStockDividendValue_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Issuance of Preferred Stock Dividend</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AccountsPayableCurrentRelatedParties_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Accounts Payable Current Related Parties</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AccountsPayableCurrentRelatedParties_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Accounts Payable Current Related Parties</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AccountsPayableCurrentRelatedParties_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accounts payable and accrued expenses - related parties</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseRollingYearTwo_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finite-Lived Intangible Assets, Amortization Expense, Rolling Year Two</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseRollingYearTwo_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">2028</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SponsorTypeDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Sponsor Type [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StatementEquityComponentsAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Equity Components [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StatementEquityComponentsAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Equity Components [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsVestedAndExpectedToVestExercisableWeightedAverageRemainingContractualTerm1_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Vested and Expected to Vest, Exercisable, Weighted Average Remaining Contractual Term</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsVestedAndExpectedToVestExercisableWeightedAverageRemainingContractualTerm1_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Weighted-average period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccountsPayableCurrentAndNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Accounts Payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccountsPayableCurrentAndNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Accounts Payable, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccountsPayableCurrentAndNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accounts payable - related parties</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NonoperatingIncomeExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Nonoperating Income (Expense)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NonoperatingIncomeExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total other expense, net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NonoperatingIncomeExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Non operating Income Expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EquityComponentDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Equity Component [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EquityComponentDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Equity Component [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TrancheObligationsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">tranche obligations</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TrancheObligationsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Tranche Obligations [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TrancheObligationsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Tranche Obligations [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfShareBasedCompensationActivityTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Payment Arrangement, Activity [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfShareBasedCompensationActivityTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Summary of stock-based compensation expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueByAssetClassAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Asset Class [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GoodwillAndIntangibleAssetsDisclosureAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Goodwill and Intangible Assets Disclosure [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredRevenueNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Deferred revenue, less current portion</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredRevenueNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Deferred Revenue, Noncurrent</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredRevenueNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Deferred Revenue, Noncurrent, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfRecognizedIdentifiedAssetsAcquiredAndLiabilitiesAssumedTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Recognized Asset Acquired and Liability Assumed [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfRecognizedIdentifiedAssetsAcquiredAndLiabilitiesAssumedTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Consideration Paid and The Fair Value of Assets Acquired, and Liabilities Assumed</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_AddressTypeDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Address Type [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongTermDebtPercentageBearingFixedInterestRate_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Long-Term Debt, Percentage Bearing Fixed Interest, Percentage Rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongTermDebtPercentageBearingFixedInterestRate_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Interest rate for term loans</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IPOMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">IPO [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IPOMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">IPO</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfCommonStockToSettlePreferredStockDividend_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Issuance of Common Stock to Settle Preferred Stock Dividend</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfCommonStockToSettlePreferredStockDividend_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Issuance of Common Stock to Settle Preferred Stock Dividend</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfCommonStockToSettlePreferredStockDividend_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Issuance of Common Stock to Settle Preferred Stock Dividend amount</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_MachineryAndEquipmentMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Machinery and Equipment [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_MachineryAndEquipmentMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Machinery and equipment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_MajorCustomersAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Customer [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_MajorCustomersAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Customer [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongTermNotesPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Notes Payable, Noncurrent</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongTermNotesPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Notes Payable, Noncurrent, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongTermNotesPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Notes payable, non-current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongTermNotesPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Notes payable, less current portion</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccountsPayableInterestBearingCurrentAndNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Accounts Payable, Interest-Bearing</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AccountsPayableInterestBearingCurrentAndNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Interest reserve accounts</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DepreciationAndAmortization_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Depreciation, Depletion and Amortization, Nonproduction</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DepreciationAndAmortization_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Depreciation, Depletion and Amortization, Nonproduction, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DepreciationAndAmortization_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Depreciation and amortization</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DepreciationAndAmortization_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Depreciation and amortization</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfUnderwritingDiscountsAndCommissionsAndOtherOfferingCostsSharesDuringPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Issuance Of Underwriting Discounts And Commissions And Other Offering Costs Shares During Period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfUnderwritingDiscountsAndCommissionsAndOtherOfferingCostsSharesDuringPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Issuance Of Underwriting Discounts And Commissions And Other Offering Costs Shares During Period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfUnderwritingDiscountsAndCommissionsAndOtherOfferingCostsSharesDuringPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Issuance Of Underwriting Discounts And Commissions And Other Offering Costs, Shares</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_RedwireCorporationMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Redwire Corporation [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_RedwireCorporationMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Redwire Corporation [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_RedwireCorporationMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Redwire Corporation</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_MaximumMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Maximum [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_MaximumMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Maximum</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfPreferredStockDividendShares_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Issuance of Preferred Stock Dividend, Sheres</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfPreferredStockDividendShares_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Issuance of Preferred Stock Dividend Shares</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IssuanceOfPreferredStockDividendShares_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Issuance of Preferred Stock Dividend Shares</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityAddressesAddressTypeAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Entity Addresses, Address Type [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LiabilitiesFairValueDisclosure_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Liabilities, Fair Value Disclosure</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LiabilitiesFairValueDisclosure_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Liabilities, Fair Value Disclosure, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LiabilitiesFairValueDisclosure_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Total financial liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_RangeAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Statistical Measurement [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_RangeAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Statistical Measurement [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsExercisesInPeriodWeightedAverageExercisePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangements by Share-Based Payment Award, Options, Exercises in Period, Weighted Average Exercise Price</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsExercisesInPeriodWeightedAverageExercisePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Weighted-Average Exercise Price, Options exercised</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAccumulatedAmortization_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finite-Lived Intangible Assets, Accumulated Amortization</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAccumulatedAmortization_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Internally-developed software not yet subject to amortization</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAccumulatedAmortization_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accumulated Amortization</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PrepaidExpenseCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Prepaid Expense, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PrepaidExpenseCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Prepaid Expense, Current, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PrepaidExpenseCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Advanced payments, current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonStockValue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Common Stock, Value, Issued</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CommonStockValue_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Common stock, $0.0001 par value, 1,000,000 and 154,397 shares authorized as of March 31, 2026 and December 31, 2025, respectively; 159,276 and 13,241 shares issued and outstanding as of March 31, 2026 and December 31, 2025, respectively</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SponsorTypeAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Sponsor Type [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SponsorTypeAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Sponsor Type [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PropertyPlantAndEquipmentMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Property, Plant and Equipment [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PropertyPlantAndEquipmentMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Property and Equipment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersDiluted_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Net Income (Loss) Available to Common Stockholders, Diluted</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersDiluted_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Net loss available to common stockholders, diluted</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FinanceLeasePrincipalPayments_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finance Lease, Principal Payments</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FinanceLeasePrincipalPayments_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Principal payments on finance leases</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsForFees_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Payments for Other Fees</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsForFees_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Annual fee</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsForFees_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">lender fees</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_WarrantsAndRightsOutstanding_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Warrants and Rights Outstanding</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_WarrantsAndRightsOutstanding_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Warrant liability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockholdersEquityIncludingPortionAttributableToNoncontrollingInterest_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Equity, Including Portion Attributable to Noncontrolling Interest</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockholdersEquityIncludingPortionAttributableToNoncontrollingInterest_lbl" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel">Equity, ending balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockholdersEquityIncludingPortionAttributableToNoncontrollingInterest_lbl" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel">Equity, beginning balance</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockholdersEquityIncludingPortionAttributableToNoncontrollingInterest_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total stockholders' equity</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SecuredOvernightFinancingRateSofrMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Secured Overnight Financing Rate (SOFR) [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockDividendsIncomeStatementImpact_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Preferred Stock Dividends, Income Statement Impact</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockDividendsIncomeStatementImpact_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Less: dividends accrued on preferred stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockDividendsIncomeStatementImpact_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Less: Accretion of dividends</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockDividendsIncomeStatementImpact_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Less: Accretion of dividends</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AmortizationOfFinancingCosts_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Amortization of Debt Issuance Costs</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AmortizationOfFinancingCosts_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Amortization of debt issuance costs</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DefaultScenarioMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Default Scenario</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DefaultScenarioMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Default Scenario [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_DefaultScenarioMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Default Scenario [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NonrecurringAdjustmentDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Pro Forma Information, Nonrecurring Adjustment [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedRightOfUseAssets_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination Recognized Identifiable Assets Acquired And Liabilities Assumed Right of Use Assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedRightOfUseAssets_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Business Combination Recognized Identifiable Assets Acquired And Liabilities Assumed Right of Use Assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedRightOfUseAssets_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Right-of-use assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedVolatilityRate_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Volatility</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedVolatilityRate_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Fair Value Assumptions, Expected Volatility Rate</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedVolatilityRate_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Expected volatility</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockSharesAuthorized_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Preferred Stock, Shares Authorized</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PreferredStockSharesAuthorized_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Preferred stock, shares authorized (in shares)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MaintainMinimumInterestReserveAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Maintain Minimum Interest Reserve Amount</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MaintainMinimumInterestReserveAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Maintain Minimum Interest Reserve Amount</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MaintainMinimumInterestReserveAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Maintain minimum interest reserve amount</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsByMajorClassAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finite-Lived Intangible Assets by Major Class [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentAnnualPrincipalPayment_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Repayment of debts</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentAnnualPrincipalPayment_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Debt Instrument, Annual Principal Payment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RelatedPartyTransactionLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Related Party Transaction [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RelatedPartyTransactionLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Related Party Transaction [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RevenueFromContractWithCustomerTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Revenue from Contract with Customer [Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RevenueFromContractWithCustomerTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Revenue</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfBusinessAcquisitionsByAcquisitionTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfBusinessAcquisitionsByAcquisitionTable_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Business Combination [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityRegistrantName_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Entity Registrant Name</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityRegistrantName_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Entity Registrant Name</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Debt Disclosure [Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtDisclosureTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Notes Payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromSaleOfShortTermInvestments_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Proceeds from Sale of Short-Term Investments</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromSaleOfShortTermInvestments_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Proceeds from sale of short-term investments</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CustomerConcentrationRiskMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Customer Concentration Risk [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CustomerConcentrationRiskMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Customer Concentration Risk [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DeferredRevenueArrangementLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Deferred Revenue Arrangement [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherNonoperatingIncomeExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Other Nonoperating Income (Expense)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherNonoperatingIncomeExpense_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Other expense, net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherNonoperatingIncomeExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Other Nonoperating Income (Expense), Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherNonoperatingIncomeExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Other (expense) income, net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherNonoperatingIncomeExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Other (expense) income, net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AdditionalPaidInCapital_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Additional Paid in Capital</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AdditionalPaidInCapital_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Additional paid-in capital, net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AccruedPayrollCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Payroll-related expenses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AccruedPayrollCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Accrued Payroll Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AccruedPayrollCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Accrued Payroll Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SeriesDPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Series D Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SeriesDPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Series D Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SubsequentEventsAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Subsequent Events [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ScheduleOfPresentsTheKeyInputsAppliedInTheValuationTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule Of Presents The Key Inputs Applied In The Valuation [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ScheduleOfPresentsTheKeyInputsAppliedInTheValuationTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Schedule Of Presents The KeyInputs Applied In The Valuation [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ScheduleOfPresentsTheKeyInputsAppliedInTheValuationTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of presents the key inputs applied in the valuation of the RPM Call Option and majority sponsortop-up</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncomeTaxExpenseBenefit_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Income Tax Expense (Benefit)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncomeTaxExpenseBenefit_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Income Tax Expense (Benefit), Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncomeTaxExpenseBenefit_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Income tax expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncomeTaxExpenseBenefit_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Provision for income taxes</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncomeTaxExpenseBenefit_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel">Income tax benefit</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncomeTaxExpenseBenefit_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Income tax expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromIssuanceOfCommonStock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Proceeds from Issuance of Common Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromIssuanceOfCommonStock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Proceeds from the issuance of Common stock, net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ProceedsFromIssuanceOfCommonStock_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Proceeds from issuance of Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NoninterestExpenseInvestmentAdvisoryFees_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Investment Advisory Fee</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NoninterestExpenseInvestmentAdvisoryFees_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Consulting and advisory services fees</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Value Measurement Inputs and Valuation Techniques [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Tables Present the Key Inputs Applied In the Valuations Warrants</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityCommonStockSharesOutstanding_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Entity Common Stock, Shares Outstanding</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ConcentrationRiskCreditRisk_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Concentration Risk, Credit Risk, Policy [Policy Text Block]</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ConcentrationRiskCreditRisk_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Allowance for Credit Losses</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LesseeOperatingLeaseRemainingLeaseTerm_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Operating Lease Remaining Term</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LesseeOperatingLeaseRemainingLeaseTerm_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Lessee, Operating Lease, Remaining Lease Term</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_ConsolidationItemsAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Consolidation Items [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="srt_ConsolidationItemsAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Consolidation Items [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_OtherRevenueMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Other Revenue [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_OtherRevenueMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Other Revenue [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_OtherRevenueMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Other Revenue</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsToAcquirePropertyPlantAndEquipment_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Payments to Acquire Property, Plant, and Equipment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsToAcquirePropertyPlantAndEquipment_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel">Purchases of property and equipment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsToAcquirePropertyPlantAndEquipment_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Purchases of property and equipment and internal-use software</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsGrantsInPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Equity Instruments Other than Options, Grants in Period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsGrantsInPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Number of shares, Granted</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DepositLiabilitiesAccruedInterest_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Deposit Liabilities, Accrued Interest</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DepositLiabilitiesAccruedInterest_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accrued interest</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_Assets_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_Assets_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ProbabilityWeightedFairValuePerCommonWarrant_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Probability Weighted Fair Value Per Common Warrant</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ProbabilityWeightedFairValuePerCommonWarrant_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Probability Weighted Fair Value Per Common Warrant</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ProbabilityWeightedFairValuePerCommonWarrant_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Total probability weighted value per Common Warrant</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_RightOfUseAssetObtainedInForLeaseLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Right Of Use Asset ObtainedIn For LeaseLiability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_RightOfUseAssetObtainedInForLeaseLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Right Of Use Asset ObtainedIn For LeaseLiability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_RightOfUseAssetObtainedInForLeaseLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Right of use For lease liability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfPropertyPlantAndEquipmentTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Property, Plant and Equipment [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfPropertyPlantAndEquipmentTable_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Property, Plant and Equipment [Table]</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SaleOfStockNumberOfSharesIssuedInTransaction_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Sale of Stock, Number of Shares Issued in Transaction</link:label>
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          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SaleOfStockNumberOfSharesIssuedInTransaction_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Number of shares sold</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseRollingYearFour_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finite-Lived Intangible Assets, Amortization Expense, Rolling Year Four</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseRollingYearFour_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">2030</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AverageFairValuePerWarrant_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Average Fair Value Per Warrant</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AverageFairValuePerWarrant_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Average Fair Value Per Warrant</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_AverageFairValuePerWarrant_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Average value per Series J Warrant</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssuedDuringPeriodSharesNewIssues_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Stock Issued During Period, Shares, New Issues</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssuedDuringPeriodSharesNewIssues_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Issuance of stock, net (in shares)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_StockIssuedDuringPeriodSharesNewIssues_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Common Stock, Initial public offering</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LeaseCostAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Lease expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LeaseCostAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Lease, Cost [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_TemporaryEquitySharesAuthorized_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Temporary Equity, Shares Authorized</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_TemporaryEquitySharesAuthorized_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Temporary equity, shares authorized (in shares)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Compensation Arrangement by Share-Based Payment Award, Equity Instruments Other than Options, Vested in Period</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriod_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Number of shares, Vested</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriod_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Number of shares, Vested</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationIntegrationRelatedCosts_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Integration-Related Cost, Expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationIntegrationRelatedCosts_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Acquisition related costs</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationsAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_RestrictedCashMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Restricted Cash [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_RestrictedCashMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Restricted Cash [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_RestrictedCashMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Restricted cash</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EarningsPerShareDiluted_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Diluted</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EarningsPerShareDiluted_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Earnings Per Share, Diluted</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EarningsPerShareDiluted_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Net loss per share attributable to common stockholders - diluted</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfSegmentReportingInformationBySegmentTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule of Segment Reporting Information, by Segment [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfSegmentReportingInformationBySegmentTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Segment Reporting Information, by Segment</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingSegmentsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Operating Segments [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingSegmentsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Operating Segments [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LandAndLandImprovementsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Land and Land Improvements [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LandAndLandImprovementsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Land and land improvements [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IPOClosingPreferredStockDividendMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">IPO Closing Preferred Stock Dividend [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IPOClosingPreferredStockDividendMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">IPO Closing Preferred Stock Dividend [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_IPOClosingPreferredStockDividendMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">IPO Closing Preferred Stock Dividend</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ClassA1PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Class A-1 Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ClassA1PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Class A-1 Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ClassA1PreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Class A-1 Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_CityAreaCode_lbl" xlink:role="http://www.xbrl.org/2003/role/label">City Area Code</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_CityAreaCode_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">City Area Code</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionsProFormaRevenue_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Pro Forma Information, Pro Forma Revenue</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionsProFormaRevenue_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Business combination post-acquisition revenue</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionsProFormaRevenue_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Pro forma revenue</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherComprehensiveIncomeLossNetOfTax_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Other Comprehensive Income (Loss), Net of Tax</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherComprehensiveIncomeLossNetOfTax_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Other Comprehensive Income (Loss), Net of Tax, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OtherComprehensiveIncomeLossNetOfTax_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Net income and comprehensive income</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FinanceLeaseLiabilityNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Finance Lease, Liability, Noncurrent</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FinanceLeaseLiabilityNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Finance lease liability, less current portion</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FinanceLeaseLiabilityNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Finance lease liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LeaseAgreementsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Lease Agreements [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LeaseAgreementsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Office building lease Member</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentFaceAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Debt Instrument, Face Amount</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtInstrumentFaceAmount_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Principal amount</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LineOfCreditFacilityMaximumBorrowingCapacity_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Line of Credit Facility, Maximum Borrowing Capacity</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LineOfCreditFacilityMaximumBorrowingCapacity_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Maximum borrowing capacity</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessAcquisitionLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Business Combination [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TermALoansMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Term A Loans [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TermALoansMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Term A Loans [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_TermALoansMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Term A Loans</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ProceedsOfUnderwritngDiscounts_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Proceeds Of Underwritng Discounts</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ProceedsOfUnderwritngDiscounts_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Proceeds Of Underwritng Discounts</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ProceedsOfUnderwritngDiscounts_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Proceeds of underwritng discounts</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfEntityWideRevenueByMajorCustomersByReportingSegmentsTable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule of Revenue by Major Customers, by Reporting Segments [Table]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GainLossRelatedToLitigationSettlement_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Gain (Loss) from Litigation Settlement</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GainLossRelatedToLitigationSettlement_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Gain (Loss) from Litigation Settlement, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_GainLossRelatedToLitigationSettlement_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Gain on settlement of contingent liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NoncashOrPartNoncashAcquisitionIntangibleAssetsAcquired1_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Noncash or Part Noncash Acquisition, Intangible Assets Acquired</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NoncashOrPartNoncashAcquisitionIntangibleAssetsAcquired1_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Acquisition of internal-use software licenses and obligations</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SegmentReportingInformationLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Segment Reporting Information [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SegmentReportingInformationLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Segment Reporting Information [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesJPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Series J Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesJPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Series J Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesJPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Series J Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LeaseCost_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Lease, Cost</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LeaseCost_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Total lease expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SpacecraftSolutionsRevenueMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Spacecraft Solutions Revenue [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SpacecraftSolutionsRevenueMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Spacecraft Solutions Revenue [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SpacecraftSolutionsRevenueMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Spacecraft Solutions Revenue</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractualObligationAnnualInstallments_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Contractual Obligation, Annual Installments</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractualObligationAnnualInstallments_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Contractual Obligation, Annual Installments</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractualObligationAnnualInstallments_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Contractual obligation, annual installments</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PropertyPlantAndEquipmentNet_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Property, Plant and Equipment, Net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PropertyPlantAndEquipmentNet_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Property and equipment, net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PropertyPlantAndEquipmentNet_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Property and equipment, net</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RightOfUseAssetObtainedInExchangeForFinanceLeaseLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Right-of-Use Asset Obtained in Exchange for Finance Lease Liability</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RightOfUseAssetObtainedInExchangeForFinanceLeaseLiability_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Right-of-use asset acquired in exchange for finance lease liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingIncomeLoss_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Operating Income (Loss)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingIncomeLoss_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Loss from operations</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingIncomeLoss_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Loss from operations</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingIncomeLoss_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Loss from operations</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_RentIncraeseAnuallyInPercent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Rent Incraese annual by Percentage</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_RentIncraeseAnuallyInPercent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Rent Incraese anually in Percent</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_RentIncraeseAnuallyInPercent_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Rent Incraese anually in Percent</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Less: Current portion</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Debt, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DebtCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Debt, Current, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RetainedEarningsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Retained Earnings [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RetainedEarningsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accumulated Deficit</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CashAndCashEquivalentsAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Cash and Cash Equivalents [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesSeedPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Series Seed Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesSeedPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Series Seed Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_SeriesSeedPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Series Seed Preferred Stock</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EmployeeBenefitsAndShareBasedCompensation_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Stock-based compensation expense.</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_EmployeeBenefitsAndShareBasedCompensation_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Employee Benefits and Share-Based Compensation</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensation_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Payment Arrangement, Noncash Expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensation_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Stock-based compensation</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensation_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Stock-based compensation expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ShareBasedCompensation_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Share-Based Payment Arrangement, Noncash Expense, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CompensationExpenseExcludingCostOfGoodAndServiceSold_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Compensation Expense, Excluding Cost of Good and Service Sold</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CompensationExpenseExcludingCostOfGoodAndServiceSold_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Compensation</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_CompensationExpenseExcludingCostOfGoodAndServiceSold_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Compensation</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingLeaseLiabilityNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Operating Lease, Liability, Noncurrent</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingLeaseLiabilityNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Operating lease liabilities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_OperatingLeaseLiabilityNoncurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Operating lease liability, less current portion</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueInputsLevel1Member_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Value, Inputs, Level 1 [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FairValueInputsLevel1Member_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Level 1</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ClassOfStockLineItems_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Class of Stock [Line Items]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DisclosureTextBlockAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Disclosure Text Block [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetCashProvidedByUsedInFinancingActivities_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Cash Provided by (Used in) Financing Activity, Including Discontinued Operation</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetCashProvidedByUsedInFinancingActivities_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Net cash (used in) provided by financing activities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NotesPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Notes Payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NotesPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Notes payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NotesPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Notes Payable, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NotesPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Notes payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LeaseCostTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Lease, Cost [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LeaseCostTableTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Lease Expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_UnamortizedDebtIssuanceExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Unamortized Debt Issuance Expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_UnamortizedDebtIssuanceExpense_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Unamortized debt issuance costs</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DevelopedTechnologyRightsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Developed Technology Rights [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DevelopedTechnologyRightsMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Developed technology</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetsWeightedAverage_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Fair Value Measurement With Unobservable Inputs Reconciliation Recurring Basis Assets Weighted Average</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetsWeightedAverage_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Fair Value Measurement With Unobservable Inputs Reconciliation Recurring Basis Assets Weighted Average</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetsWeightedAverage_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Total probability weighted average value per Series J Warrant</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Share-BasedPaymentArrangementOptionIncreaseForCostRecognitionShares_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share-Based Payment Arrangement, Option, Increase for Cost Recognition, Shares</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Share-BasedPaymentArrangementOptionIncreaseForCostRecognitionShares_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Share-Based Payment Arrangement, Option, Increase for Cost Recognition, Shares</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Share-BasedPaymentArrangementOptionIncreaseForCostRecognitionShares_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Stock-based compensation (in shares)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NonoperatingIncomeExpenseAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Nonoperating Income (Expense) [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NonoperatingIncomeExpenseAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Other income (expense):</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NonoperatingIncomeExpenseAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/verboseLabel">Other expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_DerivativeContractTypeDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Derivative Contract [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MaterialMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Material</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MaterialMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Material [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MaterialMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Material Member</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsForMortgageDeposits_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Payments for Mortgage Deposits</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaymentsForMortgageDeposits_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Payments on mortgage principal</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentAssetsReceivables_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Recognized Asset Acquired, Receivable, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentAssetsReceivables_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Accounts receivable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Schedule of Antidilutive Securities Excluded from Computation of Earnings Per Share [Table Text Block]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTextBlock_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Schedule of Antidilutive Securities Excluded from Computation of Earnings Per Share</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentAssetsMarketableSecurities_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Short-term investments</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentAssetsMarketableSecurities_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Business Combination, Recognized Asset Acquired, Investment in Debt and Equity Securities, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityInteractiveDataCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Entity Interactive Data Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityInteractiveDataCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Entity Interactive Data Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_IncomeStatementLocationDomain_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Statement of Income Location, Balance [Domain]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MaterialsAndConsumables_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Materials and Consumables</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MaterialsAndConsumables_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Materials and Consumables</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MaterialsAndConsumables_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Materials and consumables</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_MaterialsAndConsumables_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Materials and consumables</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ContractualInterestExpenseOnPrepetitionLiabilitiesNotRecognizedInStatementOfOperations_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Contractual Interest Expense on Prepetition Liabilities Not Recognized in Statement of Operations</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ContractualInterestExpenseOnPrepetitionLiabilitiesNotRecognizedInStatementOfOperations_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Contractual interest expense</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Cash Provided by (Used in) Operating Activity, Including Discontinued Operation [Abstract]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Cash flows from operating activities</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FurnitureAndFixturesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Furniture and fixtures [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_FurnitureAndFixturesMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Furniture and Fixtures [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ConvertiblePreferredStockSharesIssuedUponConversion_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Convertible Preferred Stock, Shares Issued upon Conversion</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ConvertiblePreferredStockSharesIssuedUponConversion_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Convertible preferred stock, shares issued upon conversion</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RepaymentsOfLinesOfCredit_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Repayments of Lines of Credit</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RepaymentsOfLinesOfCredit_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Repayment of Revolving Credit Facility</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaidInKindInterest_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Paid-in-Kind Interest</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_PaidInKindInterest_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Capitalized interest (paid-in-kind)</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Non-CashInventoryWrite-Off_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Non-Cash Inventory Write-Off</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Non-CashInventoryWrite-Off_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Non-Cash Inventory Write-Off</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_Non-CashInventoryWrite-Off_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Non-cash inventory write-off</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_UnbilledReceivablesCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Unbilled Receivables, Current</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_UnbilledReceivablesCurrent_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Unbilled receivables</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_InterestExpenseOnPurchaseAndLeasebackAgreement_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Interest Expense On Purchase and Leaseback Agreement</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_InterestExpenseOnPurchaseAndLeasebackAgreement_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Interest Expense On Purchase and Leaseback Agreement</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_InterestExpenseOnPurchaseAndLeasebackAgreement_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Interest expense on purchase and leaseback agreement</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ContractWithCustomerAssetNet_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Contract with Customer, Asset, after Allowance for Credit Loss</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ContractWithCustomerAssetNet_lbl" xlink:role="http://www.xbrl.org/2003/role/totalLabel">Contract with Customer, Asset, after Allowance for Credit Loss, Total</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_ContractWithCustomerAssetNet_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Contract assets</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementOptionOutstandingNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Number of units</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementOptionOutstandingNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share Based Compensation Arrangement Option Outstanding Number</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ShareBasedCompensationArrangementOptionOutstandingNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Share Based Compensation Arrangement Option Outstanding Number</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ConvertibleSeriesSeedPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Convertible Series Seed Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ConvertibleSeriesSeedPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Convertible Series Seed Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ConvertibleSeriesSeedPreferredStockMember_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Convertible Series Seed Preferred Stock [Member]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityTaxIdentificationNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Entity Tax Identification Number</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="dei_EntityTaxIdentificationNumber_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Entity Tax Identification Number</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PrepaidPurchaseCredit_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Prepaid Purchase Credit</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PrepaidPurchaseCredit_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Prepaid Purchase Credit</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_PrepaidPurchaseCredit_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Prepaid purchase credit</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SharePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Share Price</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_SharePrice_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Common Stock Price</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RepaymentsOfNotesPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Repayments of Notes Payable</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RepaymentsOfNotesPayable_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Repayment of notes payable - related parties</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_RepaymentsOfNotesPayable_lbl" xlink:role="http://www.xbrl.org/2009/role/negatedLabel">Repayment of notes payable - related parties</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongtermDebtTypeAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Long-Term Debt, Type [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_LongtermDebtTypeAxis_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Long-Term Debt, Type [Axis]</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractWithCustomerLossProvisionPeriodIncreaseDecrease_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Contract With Customer Loss Provision Period Increase Decrease</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractWithCustomerLossProvisionPeriodIncreaseDecrease_lbl" xlink:role="http://www.xbrl.org/2003/role/documentation">Contract With Customer Loss Provision Period Increase Decrease</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="fly_ContractWithCustomerLossProvisionPeriodIncreaseDecrease_lbl" xlink:role="http://www.xbrl.org/2003/role/terseLabel">Contract With Customer Loss Provision Period Increase Decrease</link:label>
          <link:label xml:lang="en-US" xlink:type="resource" xlink:label="us-gaap_AdjustmentsToAdditionalPaidInCapitalIncreaseInCarryingAmountOfRedeemablePreferredStock_lbl" xlink:role="http://www.xbrl.org/2003/role/label">Adjustments to Additional Paid in Capital, Increase in Carrying Amount of Redeemable Preferred Stock</link:label>
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  <xsd:element id="fly_BusinessCombinationCashConsiderationPaid" name="BusinessCombinationCashConsiderationPaid" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_ExerciseOfCommonWarrantsIntoCommonStock" name="ExerciseOfCommonWarrantsIntoCommonStock" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_PaymentForPurchaseAndLeasebackAgreements" name="PaymentForPurchaseAndLeasebackAgreements" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_AccuredInterestFromSubordinatedConvertiblePromissoryNotes" name="AccuredInterestFromSubordinatedConvertiblePromissoryNotes" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_Rule10B51ArrModifiedFlag" name="Rule10B51ArrModifiedFlag" type="xbrli:booleanItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_OtherRevenueMember" name="OtherRevenueMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_PreferredStockIssuedAtAPurchasePrice" name="PreferredStockIssuedAtAPurchasePrice" type="dtr-types:perShareItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_InterestExpenseContractual" name="InterestExpenseContractual" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_ConsultingAgreementMember" name="ConsultingAgreementMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAccruedExpenses" name="BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAccruedExpenses" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetsWeightedAverage" name="FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetsWeightedAverage" type="dtr-types:perShareItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_SevenPointSeventyOnePercentDueTwoThousandTwentyNineMember" name="SevenPointSeventyOnePercentDueTwoThousandTwentyNineMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_TwoThousandSeventeenEquityIncentivePlanMember" name="TwoThousandSeventeenEquityIncentivePlanMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_Internally-DevelopedSoftwareNotYetSubjectToAmortization" name="Internally-DevelopedSoftwareNotYetSubjectToAmortization" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_MaterialMember" name="MaterialMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_AccruedPayrollCurrent" name="AccruedPayrollCurrent" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_RentIncraeseAnuallyInPercent" name="RentIncraeseAnuallyInPercent" type="dtr-types:percentItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_SciTecMember" name="SciTecMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_TransactionRelatedExpenses" name="TransactionRelatedExpenses" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_MinimumCashBalanceForPurchaseAndLeasebackAgreements" name="MinimumCashBalanceForPurchaseAndLeasebackAgreements" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_ConvertibleNotesMember" name="ConvertibleNotesMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_PercentageOfNonCompoundingRate" name="PercentageOfNonCompoundingRate" type="xbrli:pureItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_DeemedSharesIssuedPricePerShare" name="DeemedSharesIssuedPricePerShare" type="dtr-types:perShareItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_Non-CashInventoryWrite-Off" name="Non-CashInventoryWrite-Off" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_ContractWithCustomerLossProvisionPeriodIncreaseDecrease" name="ContractWithCustomerLossProvisionPeriodIncreaseDecrease" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_DiscountForLackOfMarketability" name="DiscountForLackOfMarketability" type="dtr-types:percentItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_NonCancelableServiceContractCommitmentsMember" name="NonCancelableServiceContractCommitmentsMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_IssuanceOfUnderwritingDiscountsAndCommissionsAndOtherOfferingCostsSharesDuringPeriod" name="IssuanceOfUnderwritingDiscountsAndCommissionsAndOtherOfferingCostsSharesDuringPeriod" type="xbrli:sharesItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_DefaultScenarioMember" name="DefaultScenarioMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_SeriesJPreferredStockWarrantsMember" name="SeriesJPreferredStockWarrantsMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_ShareBasedCompensationArrangementOptionOutstandingNumber" name="ShareBasedCompensationArrangementOptionOutstandingNumber" type="xbrli:sharesItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_InternalUseSoftwareMember" name="InternalUseSoftwareMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_EquityMethodInvestmentMember" name="EquityMethodInvestmentMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_CustomerFourMember" name="CustomerFourMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_SeriesJWarrantMember" name="SeriesJWarrantMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_Share-BasedCompensationArrangementsByShare-BasedPaymentAwardOptionsCanceledInPeriodWeightedAverageExercisePrice" name="Share-BasedCompensationArrangementsByShare-BasedPaymentAwardOptionsCanceledInPeriodWeightedAverageExercisePrice" type="dtr-types:perShareItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_AverageFairValuePerWarrant" name="AverageFairValuePerWarrant" type="dtr-types:perShareItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_NumberOfReservedSharesIncreaseApproximately" name="NumberOfReservedSharesIncreaseApproximately" type="xbrli:sharesItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_ShareBasedCompensationArrangementOptionOutstandingFairValueMajoritySponsor" name="ShareBasedCompensationArrangementOptionOutstandingFairValueMajoritySponsor" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_ProbabilityWeightedFairValuePerCommonWarrant" name="ProbabilityWeightedFairValuePerCommonWarrant" type="dtr-types:perShareItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_AssetsAndLiabilitiesLesseeTableTextBlock" name="AssetsAndLiabilitiesLesseeTableTextBlock" type="dtr-types:textBlockItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_CommonStockWarrantsMember" name="CommonStockWarrantsMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_ITRelatedEquipmentMember" name="ITRelatedEquipmentMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_Internal-UseSoftwareAndInternally-DevelopedSoftwareMember" name="Internal-UseSoftwareAndInternally-DevelopedSoftwareMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_SecuredOverFinancingRateMember" name="SecuredOverFinancingRateMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_WarrantsToPurchaseSeriesJPreferredStockMember" name="WarrantsToPurchaseSeriesJPreferredStockMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_SwinglineLoans" name="SwinglineLoans" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_IPOScenarioMember" name="IPOScenarioMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_SMINoteMember" name="SMINoteMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_NumberOfReservedSharesIncrease" name="NumberOfReservedSharesIncrease" type="xbrli:sharesItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_ShareBasedCompensationArrangementOptionFairValue" name="ShareBasedCompensationArrangementOptionFairValue" type="dtr-types:perShareItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_EightPointThirtyOnePercentDueTwoThousandThirtyOneMember" name="EightPointThirtyOnePercentDueTwoThousandThirtyOneMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_ChangeOfControlScenarioMember" name="ChangeOfControlScenarioMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_ShareBasedCompensationArrangementByShare-BasedPaymentAwardEquityInstrumentsOtherThanOptionsCancelledInPeriod" name="ShareBasedCompensationArrangementByShare-BasedPaymentAwardEquityInstrumentsOtherThanOptionsCancelledInPeriod" type="xbrli:sharesItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_MinimumLiquidityMaintenanceCovenant" name="MinimumLiquidityMaintenanceCovenant" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_IssuanceOfCommonStockToSettlePreferredStockDividend" name="IssuanceOfCommonStockToSettlePreferredStockDividend" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_ConvertibleSeriesSeed-1PreferredStockMember" name="ConvertibleSeriesSeed-1PreferredStockMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_ChangeInRight-Of-UseAssets" name="ChangeInRight-Of-UseAssets" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_ExitFeePercentage" name="ExitFeePercentage" type="dtr-types:percentItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_ScheduleOfPresentsTheKeyInputsAppliedInTheValuationTableTextBlock" name="ScheduleOfPresentsTheKeyInputsAppliedInTheValuationTableTextBlock" type="dtr-types:textBlockItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_ChangeInFairValueOfWarrantLiability" name="ChangeInFairValueOfWarrantLiability" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_LeaseRemainingLeaseTerm" name="LeaseRemainingLeaseTerm" type="xbrli:durationItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_AccumulatedDeficitMember" name="AccumulatedDeficitMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_NonCancelableSubscriptionBasedServicesMember" name="NonCancelableSubscriptionBasedServicesMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_BusinessCombinationTotalConsiderationPaid" name="BusinessCombinationTotalConsiderationPaid" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_PreferredStockConvertedToCommonSharesMember" name="PreferredStockConvertedToCommonSharesMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_InterestExpenseOnPurchaseAndLeasebackAgreement" name="InterestExpenseOnPurchaseAndLeasebackAgreement" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_ProceedsFromNotesPayableRelatedParties" name="ProceedsFromNotesPayableRelatedParties" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_OtherNotesMember" name="OtherNotesMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_OtherArrangementsMember" name="OtherArrangementsMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_ScheduleOfRedeemableConvertiblePreferredStock" name="ScheduleOfRedeemableConvertiblePreferredStock" type="dtr-types:textBlockItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_GainOnSettlementOfContingentLiabilities" name="GainOnSettlementOfContingentLiabilities" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_ProceedsOfUnderwritngDiscounts" name="ProceedsOfUnderwritngDiscounts" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_IssuanceOfDebtInExchangeOfSoftwareLicenses" name="IssuanceOfDebtInExchangeOfSoftwareLicenses" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_SpacecraftSolutionsRevenueMember" name="SpacecraftSolutionsRevenueMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_SeriesJWarrantsMember" name="SeriesJWarrantsMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_PricePerSquareFoot" name="PricePerSquareFoot" type="dtr-types:areaItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_UnauditedProFormaInformationMember" name="UnauditedProFormaInformationMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_CashProceeds" name="CashProceeds" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_ShareBasedCompensationArrangementMaximumValueRecognized" name="ShareBasedCompensationArrangementMaximumValueRecognized" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_FinancingAgreementMember" name="FinancingAgreementMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_SeriesMPreferredStockMember" name="SeriesMPreferredStockMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedRightOfUseAssets" name="BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedRightOfUseAssets" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_EventProbabilityPercentage" name="EventProbabilityPercentage" type="dtr-types:percentItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_IssuanceOfPreferredStockDividendShares" name="IssuanceOfPreferredStockDividendShares" type="xbrli:sharesItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_FireflyMember" name="FireflyMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_PercentageOfNonAccruingOrignalIssuePricePerShare" name="PercentageOfNonAccruingOrignalIssuePricePerShare" type="dtr-types:percentItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_ConvertibleSeriesSeedPreferredStockMember" name="ConvertibleSeriesSeedPreferredStockMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_WeightedAverageFairValuePerWarrant" name="WeightedAverageFairValuePerWarrant" type="dtr-types:perShareItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_TwoThousandTwentyFiveEmployeeStockPurchasePlanMember" name="TwoThousandTwentyFiveEmployeeStockPurchasePlanMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_AEOperatingMember" name="AEOperatingMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_NonRule10B51ArrModifiedFlag" name="NonRule10B51ArrModifiedFlag" type="xbrli:booleanItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_BusinessCombinationEquityConsiderationPaid" name="BusinessCombinationEquityConsiderationPaid" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_ProceedsFromRepaymentOfEmployeeNote" name="ProceedsFromRepaymentOfEmployeeNote" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_TermBLoansMember" name="TermBLoansMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_NetWorkingCapitalAdjustment" name="NetWorkingCapitalAdjustment" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_Share-BasedCompensationArrangementByShare-BasedPaymentAwardOptionsAvailableForGrantGrantsInPeriod" name="Share-BasedCompensationArrangementByShare-BasedPaymentAwardOptionsAvailableForGrantGrantsInPeriod" type="xbrli:sharesItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_SeriesD-2PreferredStockMember" name="SeriesD-2PreferredStockMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_DiscountsOnUnderwriting" name="DiscountsOnUnderwriting" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_ContractualObligationAnnualInstallments" name="ContractualObligationAnnualInstallments" type="xbrli:integerItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_CustomerMember" name="CustomerMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_DebtInstrumentCarryingAmountNetOfIssuanceCosts" name="DebtInstrumentCarryingAmountNetOfIssuanceCosts" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_LeasesTextBlock" name="LeasesTextBlock" type="dtr-types:textBlockItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_ExerciseOfCommonWarrantsIntoCommonStockShares" name="ExerciseOfCommonWarrantsIntoCommonStockShares" type="xbrli:sharesItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_NonCashInterestIncome" name="NonCashInterestIncome" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_TestStandsMember" name="TestStandsMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_FairValuePerShareNonMarketableSecurity" name="FairValuePerShareNonMarketableSecurity" type="dtr-types:perShareItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_OneTimeCostsRelatedToTheIPO" name="OneTimeCostsRelatedToTheIPO" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_MaintainMinimumInterestReserveAmount" name="MaintainMinimumInterestReserveAmount" type="xbrli:pureItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_CommonStockSharesMember" name="CommonStockSharesMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_ContractorsAndOutsideServices" name="ContractorsAndOutsideServices" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_PaymentsOnNotesPayable" name="PaymentsOnNotesPayable" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_SpecificAdditionalInterestRate" name="SpecificAdditionalInterestRate" type="dtr-types:percentItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_SeriesSeedPreferredStockMember" name="SeriesSeedPreferredStockMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_SponsorTypeAxis" name="SponsorTypeAxis" type="xbrli:stringItemType" substitutionGroup="xbrldt:dimensionItem" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_Share-BasedPaymentArrangementOptionIncreaseForCostRecognitionShares" name="Share-BasedPaymentArrangementOptionIncreaseForCostRecognitionShares" type="xbrli:sharesItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_ClassA1PreferredStockMember" name="ClassA1PreferredStockMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_IssuanceOfPreferredStockDividend" name="IssuanceOfPreferredStockDividend" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_RPMCallOptionMember" name="RPMCallOptionMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_PercentageOfNonAccruingDividendDeemedIssuePricePerShare" name="PercentageOfNonAccruingDividendDeemedIssuePricePerShare" type="dtr-types:percentItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_VotingPowerPercentage" name="VotingPowerPercentage" type="dtr-types:percentItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_CustomerOneMember" name="CustomerOneMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_SponsorTypeDomain" name="SponsorTypeDomain" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedOperatingLeaseLiabilitiesNonCurrent" name="BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedOperatingLeaseLiabilitiesNonCurrent" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_ContractualObligationNumberOfLaunches" name="ContractualObligationNumberOfLaunches" type="xbrli:integerItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_SeriesD-3PreferredStockMember" name="SeriesD-3PreferredStockMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_SpaceflightMember" name="SpaceflightMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_PrepaidPurchaseCredit" name="PrepaidPurchaseCredit" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_BelcanLLCMember" name="BelcanLLCMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_IssuanceOfUnderwritingDiscountsAndCommissionsAndOtherOfferingCostsValueDuringPeriod" name="IssuanceOfUnderwritingDiscountsAndCommissionsAndOtherOfferingCostsValueDuringPeriod" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_SeriesSeed-1PreferredStockMember" name="SeriesSeed-1PreferredStockMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_NetWorkingCapitalAdjustmentFromBusinessCombinations" name="NetWorkingCapitalAdjustmentFromBusinessCombinations" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_RedwireCorporationMember" name="RedwireCorporationMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_BusinessCombinationIncurredCost" name="BusinessCombinationIncurredCost" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_FairValuePerShareMarketableSecurity" name="FairValuePerShareMarketableSecurity" type="dtr-types:perShareItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_TermALoansMember" name="TermALoansMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_AccountsPayableCurrentRelatedParties" name="AccountsPayableCurrentRelatedParties" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_TwoThousandTwentyFiveOmnibusIncentivePlanMember" name="TwoThousandTwentyFiveOmnibusIncentivePlanMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_FairMarketValueOfSharePercentage" name="FairMarketValueOfSharePercentage" type="dtr-types:percentItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_Non-CashInvestingAndFinancingActivitiesEquityIssuedForBusinessCombination" name="Non-CashInvestingAndFinancingActivitiesEquityIssuedForBusinessCombination" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_Non-CashInterestExpense" name="Non-CashInterestExpense" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_Internally-DevelopedSoftwareMember" name="Internally-DevelopedSoftwareMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_SeriesD-1PreferredStockMember" name="SeriesD-1PreferredStockMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_WeightedAveragePricePerShare" name="WeightedAveragePricePerShare" type="dtr-types:perShareItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_LaunchRevenueMember" name="LaunchRevenueMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_Weighted-AverageRemainingContractualTermUnvested" name="Weighted-AverageRemainingContractualTermUnvested" type="xbrli:durationItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_CommonStockOutstandingCommonWarrantsExercised" name="CommonStockOutstandingCommonWarrantsExercised" type="xbrli:sharesItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_SharesOutstandingPercentage" name="SharesOutstandingPercentage" type="dtr-types:percentItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_AverageValuePerUnitAsPerMonteCarloSimulations" name="AverageValuePerUnitAsPerMonteCarloSimulations" type="xbrli:stringItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_MaterialsAndConsumables" name="MaterialsAndConsumables" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_ThirteenPointEightHundredSeventyFivePercentDueTwoThousandTwentyEightMember" name="ThirteenPointEightHundredSeventyFivePercentDueTwoThousandTwentyEightMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_IssuanceOfCommonStockToSettlePreferredStockDividendShares" name="IssuanceOfCommonStockToSettlePreferredStockDividendShares" type="xbrli:sharesItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_IncreaseDecreaseAccountsPayableRelatedParties" name="IncreaseDecreaseAccountsPayableRelatedParties" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_FollowOnOpportunitiesMember" name="FollowOnOpportunitiesMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_IssuanceOfPreferredStockDividendValue" name="IssuanceOfPreferredStockDividendValue" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_PaymentsForTaxesRelatedToNetShareSettlementOfEquityAwards" name="PaymentsForTaxesRelatedToNetShareSettlementOfEquityAwards" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_CustomerThreeMember" name="CustomerThreeMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_SeriesD-4PreferredStockMember" name="SeriesD-4PreferredStockMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_BacklogMember" name="BacklogMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_FairValueInputsDiscountRate" name="FairValueInputsDiscountRate" type="dtr-types:percentItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNetWorkingCapitalAdjustment" name="BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNetWorkingCapitalAdjustment" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_RightOfUseAssetObtainedInForLeaseLiability" name="RightOfUseAssetObtainedInForLeaseLiability" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="credit"/>
  <xsd:element id="fly_ContractWithCustomerLossProvision" name="ContractWithCustomerLossProvision" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_ROUAssetsExchangeOfLeaseLiabilities" name="ROUAssetsExchangeOfLeaseLiabilities" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_MajoritySponsorMember" name="MajoritySponsorMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true"/>
  <xsd:element id="fly_ShareBasedCompensationArrangementOptionFairValueOutstandingNumber" name="ShareBasedCompensationArrangementOptionFairValueOutstandingNumber" type="xbrli:sharesItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_NumberOfShareSold" name="NumberOfShareSold" type="xbrli:sharesItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
  <xsd:element id="fly_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsAvailableForGrantExercisedInPeriod" name="ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsAvailableForGrantExercisedInPeriod" type="xbrli:sharesItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false"/>
  <xsd:element id="fly_OperatingLeaseRight-Of-UseAssetDecreased" name="OperatingLeaseRight-Of-UseAssetDecreased" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="debit"/>
  <xsd:element id="fly_ContractualObligationCancellationFeePercentage" name="ContractualObligationCancellationFeePercentage" type="dtr-types:percentItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false"/>
</xsd:schema>
