A search for flavour-changing neutral-current processes in top-quark decays is presented. Data collected with the ATLAS detector from proton-proton collisions at the Large Hadron Collider at a centre-of-mass energy of s=13 TeV, corresponding to an integrated luminosity of 36.1 fb−1, are analysed. The search is performed using top-quark pair events, with one top quark decaying through the t → qZ (q = u, c) flavour-changing neutral-current channel, and the other through the dominant Standard Model mode t → bW. Only Z boson decays into charged leptons and leptonic W boson decays are considered as signal. Consequently, the final-state topology is characterized by the presence of three isolated charged leptons (electrons or muons), at least two jets, one of the jets originating from a b-quark, and missing transverse momentum from the undetected neutrino. The data are consistent with Standard Model background contributions, and at 95% confidence level the search sets observed (expected) upper limits of 1.7 × 10−4 (2.4 × 10−4) on the t → uZ branching ratio and 2.4 × 10−4 (3.2 × 10−4) on the t → cZ branching ratio, constituting the most stringent limits to date.[Figure not available: see fulltext.].