  Cross-sections for the production of a $Z$ boson in association with two
photons are measured in proton$-$proton collisions at a centre-of-mass energy
of 13 TeV. The data used correspond to an integrated luminosity of 139
fb$^{-1}$ recorded by the ATLAS experiment during Run 2 of the LHC. The
measurements use the electron and muon decay channels of the $Z$ boson, and a
fiducial phase-space region where the photons are not radiated from the
leptons. The integrated $Z(\rightarrow\ell\ell)\gamma\gamma$ cross-section is
measured with a precision of 12% and differential cross-sections are measured
as a function of six kinematic variables of the $Z\gamma\gamma$ system. The
data are compared with predictions from MC event generators which are accurate
to up to next-to-leading order in QCD. The cross-section measurements are used
to set limits on the coupling strengths of dimension-8 operators in the
framework of an effective field theory.
