<?xml version="1.0"?>
<ownershipDocument>

    <schemaVersion>X0508</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2026-02-24</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0001562463</issuerCik>
        <issuerName>First Internet Bancorp</issuerName>
        <issuerTradingSymbol>INBK</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001977043</rptOwnerCik>
            <rptOwnerName>Fenech Joseph A.</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>8701 E. 116TH STREET</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>FISHERS</rptOwnerCity>
            <rptOwnerState>IN</rptOwnerState>
            <rptOwnerZipCode>46038</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>1</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
        </reportingOwnerRelationship>
    </reportingOwner>

    <aff10b5One>0</aff10b5One>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2026-02-24</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>P</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>1000</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>19.83</value>
                    <footnoteId id="F1"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>10102</value>
                    <footnoteId id="F2"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>4050</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>Shares owned by GenOpp Financial Fund LP</value>
                    <footnoteId id="F3"/>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
    </nonDerivativeTable>

    <derivativeTable></derivativeTable>

    <footnotes>
        <footnote id="F1">Reflects the weighted-average price of shares purchased by the reporting person in multiple transactions at prices ranging from $19.83 to $19.87 per share. The reporting person undertakes to provide to the issuer, a security holder of the issuer or the U.S. Securities and Exchange Commission staff, upon request, full information regarding the number of shares purchased at each separate price.</footnote>
        <footnote id="F2">Includes 51 shares acquired between July 16, 2025 and January 16, 2026 pursuant to the First Internet Bancorp Dividend Reinvestment and Stock Purchase Plan.</footnote>
        <footnote id="F3">All shares are owned by GenOpp Financial Fund LP (the 'Fund'). Mr. Fenech is the indirect owner of the general partner and investment manager, respectively, of the Fund. This filing is not an admission that Mr. Fenech beneficially owns the securities covered by this report.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ Kenneth J. Lovik, Attorney-in-Fact for Joe Fenech (power of attorney previously filed)</signatureName>
        <signatureDate>2026-02-26</signatureDate>
    </ownerSignature>
</ownershipDocument>
