A search for supersymmetry targeting the direct production of winos and higgsinos is conducted in final states with either two leptons (e or mu) with the same electric charge, or three leptons. The analysis uses 139 fb(-1) of pp collision data at root s = 13TeV collected with the ATLAS detector during Run 2 of the Large Hadron Collider. No significant excess over the Standard Model expectation is observed. Simplified and complete models with and without R-parity conservation are considered. In topologies with intermediate states including either Wh or WZ pairs, wino masses up to 525 GeV and 250 GeV are excluded, respectively, for a bino of vanishing mass. Higgsino masses smaller than 440 GeV are excluded in a natural R-parity-violating model with bilinear terms. Upper limits on the production cross section of generic events beyond the Standard Model as low as 40 ab are obtained in signal regions optimised for these models and also for an R-parity-violating scenario with baryon-number-violating higgsino decays into top quarks and jets. The analysis significantly improves sensitivity to supersymmetric models and other processes beyond the Standard Model that may contribute to the considered final states.