A search for the associated production of a top-quark pair with the Higgs boson (ttH ¯) in multilepton final states is presented. The search is based on a dataset of proton-proton collisions at vs = 13 TeV and an integrated luminosity of 80 fb-1 recorded with the ATLAS detector at the CERN Large Hadron Collider. A synopsis of the final state with two same-charge light leptons (e or µ) and one hadronically-decaying t is described in more detail. Non-prompt light leptons background is estimated from simulation, with data-driven corrections and fake thad backgrounds are estimated using data-driven techniques. An excess of events consistent with ttH ¯ production, over the expected background from Standard Model processes, is found with an observed significance of 1.8 standard deviations, compared to an expectation of 3.1 standard deviations. Assuming Standard Model branching fractions, the best-fit value of the ttH ¯ production cross section is sttH ¯ = 294+-182162 fb, which is consistent with the Standard Model prediction.