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.