Abstract A search is presented for four-top-quark production using an integrated luminosity of 139 fb $$^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow/> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> of proton–proton collision data at a centre-of-mass energy of $$13~\text {TeV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>13</mml:mn> <mml:mspace/> <mml:mtext>TeV</mml:mtext> </mml:mrow> </mml:math> collected by the ATLAS detector at the LHC. Events are selected if they contain a same-sign lepton pair or at least three leptons (electrons or muons). Jet multiplicity, jet flavour and event kinematics are used to separate signal from the background through a multivariate discriminant, and dedicated control regions are used to constrain the dominant backgrounds. The four-top-quark production cross section is measured to be $$24^{+7}_{-6}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mn>24</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> fb. This corresponds to an observed (expected) significance with respect to the background-only hypothesis of 4.3 (2.4) standard deviations and provides evidence for this process.