  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 $Z(\rightarrow \! \! \nu \nu) \, t + X$ decay channel. LHC pp collision
data at a centre-of-mass energy of $\sqrt{s}=13$ TeV recorded by the ATLAS
detector in 2015 and 2016 are used, corresponding to an integrated luminosity
of 36.1 $\mathrm{fb}^{-1}$. 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.
