Measurements of inclusive and differential production cross-sections of a top-quark-top-antiquark pair in association with a W boson (tt¯W) are presented. They are performed by targeting final states with two same-sign or three isolated leptons (electrons or muons) and are based on s = 13 TeV proton-proton collision data with an integrated luminosity of 140 fb−1, recorded from 2015 to 2018 with the ATLAS detector at the Large Hadron Collider. The inclusive tt¯W production cross-section is measured to be 880 ± 80 fb, compared to a reference theoretical prediction of 745 ± 50 (scale) ± 13 (2-loop approx.) ± 19 (PDF, αs) fb. Differential cross-section measurements characterise this process in detail for the first time. Several particle-level observables are compared with a variety of theoretical predictions, which generally agree well with the normalised differential cross-section results. Additionally, the relative charge asymmetry of tt¯W+ and tt¯W− is measured inclusively to be ACrel = 0.33 ± 0.05, in very good agreement with the theoretical prediction of 0.322 ± 0.003 (scale) ± 0.007 (PDF), as well as differentially.