A search for the associated production of the Higgs boson with a top quark pair ($t\bar t H$) in multilepton final states using a dataset corresponding to an integrated luminosity of 36.1 fb$^{-1}$ of proton--proton collision data recorded by the ATLAS experiment at a centre-of-mass energy $\sqrt{s} = 13$ TeV at the Large Hadron Collider is reported. Higgs boson decays to $WW^*$, $\tau\tau$, and $ZZ^*$ are targeted. Seven final states, categorised by the number and flavour of charged-lepton candidates, are examined for the presence of the Standard Model Higgs boson with a mass of 125 GeV and a pair of top quarks. An excess of events over the expected background from other Standard Model processes is found with an observed significance of 4.1 standard deviations, compared to an expectation of 2.8 standard deviations. The best fit for the $t\bar t H$ production cross section is $\sigma(t\bar t H) = 790^{+230}_{-210}$ fb, in agreement with the Standard Model prediction of $507^{+35}_{-50}$~fb. The combination of this result with other $t\bar t H$ searches from the ATLAS experiment using the Higgs boson decay modes to $b\bar b$, $\gamma\gamma$ and $ZZ^* \to 4\ell$, has an observed significance of 4.2 standard deviations, compared to an expectation of 3.8 standard deviations. This provides evidence for the $t\bar t H$ production mode.

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