  This letter presents a search for narrow, high-mass resonances in the
$Z\gamma$ final state with the $Z$ boson decaying into a pair of electrons or
muons. The $\sqrt{s}=13$ TeV $pp$ collision data were recorded by the ATLAS
detector at the CERN Large Hadron Collider and have an integrated luminosity of
140 fb$^{-1}$. The data are found to be in agreement with the Standard Model
background expectation. Upper limits are set on the resonance production cross
section times the decay branching ratio into $Z\gamma$. For spin-0 resonances
produced via gluon-gluon fusion, the observed limits at 95% confidence level
vary between 65.5 fb and 0.6 fb, while for spin-2 resonances produced via
gluon-gluon fusion (or quark-antiquark initial states) limits vary between 77.4
(76.1) fb and 0.6 (0.5) fb, for the mass range from 220 GeV to 3400 GeV.
