A search for the leptonic charge asymmetry (Acℓ) of top-quark-antiquark pair production in association with a W boson (tt¯ W) is presented. The search is performed using final states with exactly three charged light leptons (electrons or muons) and is based on s = 13 TeV proton-proton collision data collected with the ATLAS detector at the Large Hadron Collider at CERN during the years 2015–2018, corresponding to an integrated luminosity of 139 fb ⁻¹. A profile-likelihood fit to the event yields in multiple regions corresponding to positive and negative differences between the pseudorapidities of the charged leptons from top-quark and top-antiquark decays is used to extract the charge asymmetry. At reconstruction level, the asymmetry is found to be −0.12 ± 0.14 (stat.) ± 0.05 (syst.). An unfolding procedure is applied to convert the result at reconstruction level into a charge-asymmetry value in a fiducial volume at particle level with the result of −0.11 ± 0.17 (stat.) ± 0.05 (syst.). The Standard Model expectations for these two observables are calculated using Monte Carlo simulations with next-to-leading-order plus parton shower precision in quantum chromodynamics and including next-to-leading-order electroweak corrections. They are −0.084−0.003+0.005 (scale) ± 0.006 (MC stat.) and −0.063−0.004+0.007 (scale) ± 0.004 (MC stat.) respectively, and in agreement with the measurements. [Figure not available: see fulltext.].