A search for the Zγ decay of the Higgs boson, with Z boson decays into pairs of electrons or muons is presented. The analysis uses proton–proton collision data at s = 13 TeV corresponding to an integrated luminosity of 139 fb−1 recorded by the ATLAS detector at the Large Hadron Collider. The observed data are consistent with the expected background with a p-value of 1.3%. An upper limit at 95% confidence level on the production cross-section times the branching ratio for pp→H→Zγ is set at 3.6 times the Standard Model prediction while 2.6 times is expected in the presence of the Standard Model Higgs boson. The best-fit value for the signal yield normalised to the Standard Model prediction is 2.0−0.9+1.0 where the statistical component of the uncertainty is dominant.