Measurements of both the inclusive and differential production cross sections of a top-quark-top-antiquark pair in association with a Z boson (tt¯Z) are presented. Final states with two, three or four isolated leptons (electrons or muons) are targeted. The measurements use the data recorded by the ATLAS detector in pp collisions at s = 13 TeV at the Large Hadron Collider during the years 2015–2018, corresponding to an integrated luminosity of 140 fb−1. The inclusive cross section is measured to be σtt¯Z = 0.86 ± 0.04 (stat.) ± 0.04 (syst.) pb and found to be in agreement with the most advanced Standard Model predictions. The differential measurements are presented as a function of a number of observables that probe the kinematics of the tt¯Z system. Both the absolute and normalised differential cross-section measurements are performed at particle level and parton level for specific fiducial volumes, and are compared with NLO+NNLL theoretical predictions. The results are interpreted in the framework of Standard Model effective field theory and used to set limits on a large number of dimension-6 operators involving the top quark. The first measurement of spin correlations in tt¯Z events is presented: the results are in agreement with the Standard Model expectations, and the null hypothesis of no spin correlations is disfavoured with a significance of 1.8 standard deviations.