The production of Z bosons in association with a high-energy photon (Zγ production) is studied in the neutrino decay channel of the Z boson using pp collisions at s=13 TeV. The analysis uses a data sample with an integrated luminosity of 36.1fb−1 collected by the ATLAS detector at the LHC in 2015 and 2016. Candidate Zγ events with invisible decays of the Z boson are selected by requiring significant transverse momentum (pT) of the dineutrino system in conjunction with a single isolated photon with large transverse energy (ET). The rate of Zγ production is measured as a function of photon ET, dineutrino system pT and jet multiplicity. Evidence of anomalous triple gauge-boson couplings is sought in Zγ production with photon ET greater than 600 GeV. No excess is observed relative to the Standard Model expectation, and upper limits are set on the strength of ZZγ and Zγγ couplings.[Figure not available: see fulltext.].