Measurements of inclusive and differential production cross-sections of a top-quark-top-antiquark pair in association with a W boson ($ t\overline{t}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 $ \sqrt{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 $ t\overline{t}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, α$_{s}$) 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 $ t\overline{t}{W}^{+} $ and $ t\overline{t}{W}^{-} $ is measured inclusively to be $ {A}_{\textrm{C}}^{\textrm{rel}} $ = 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.[graphic not available: see fulltext]