Measurements of both the inclusive and differential production cross sections of a top-quark-antiquark pair in association with a $Z$ boson ($tt¯Z$) are presented. The measurements are performed by targeting final states with three or four isolated leptons (electrons or muons) and are based on $s=13$ TeV proton-proton collision data with an integrated luminosity of 139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$, recorded from 2015 to 2018 with the ATLAS detector at the CERN Large Hadron Collider. The inclusive cross section is measured to be $\sigma tt¯Z=0.99\pm 0.05$ (stat.) $\pm 0.08$ (syst.) pb, in agreement with the most precise theoretical predictions. The differential measurements are presented as a function of a number of kinematic variables which probe the kinematics of the $tt¯Z$ system. Both absolute and normalised differential cross-section measurements are performed at particle and parton levels for specific fiducial volumes and are compared with theoretical predictions at different levels of precision, based on a $\chi 2/$ndf and $p$-value computation. Overall, good agreement is observed between the unfolded data and the predictions.