The results of a search for vector-like top quarks using events with exactly one lepton, at least four jets, and large missing transverse momentum are reported. The search is optimised for pair production of vector-like top quarks in the Z() t + X decay channel. LHC pp collision data at a centre-of-mass energy of s=13 TeV recorded by the ATLAS detector in 2015 and 2016 are used, corresponding to an integrated luminosity of 36.1 fb−1. No significant excess over the Standard Model expectation is seen and upper limits on the production cross-section of a vector-like T quark pair as a function of the T quark mass are derived. The observed (expected) 95% CL lower limits on the T mass are 870 GeV (890 GeV) for the weak-isospin singlet model, 1.05 TeV (1.06 TeV) for the weak-isospin doublet model and 1.16 TeV (1.17 TeV) for the pure Zt decay mode. Limits are also set on the mass as a function of the decay branching ratios, excluding large parts of the parameter space for masses below 1 TeV.[Figure not available: see fulltext.].