A search for chargino–neutralino pair production in three-lepton final states with missing transverse momentum is presented. The study is based on a dataset of s=13 TeV pp collisions recorded with the ATLAS detector at the LHC, corresponding to an integrated luminosity of 139 fb ^{- 1}. No significant excess relative to the Standard Model predictions is found in data. The results are interpreted in simplified models of supersymmetry, and statistically combined with results from a previous ATLAS search for compressed spectra in two-lepton final states. Various scenarios for the production and decay of charginos (χ~1±) and neutralinos (χ~20) are considered. For pure higgsino χ~1±χ~20 pair-production scenarios, exclusion limits at 95% confidence level are set on χ~20 masses up to 210 GeV. Limits are also set for pure wino χ~1±χ~20 pair production, on χ~20 masses up to 640 GeV for decays via on-shell W and Z bosons, up to 300 GeV for decays via off-shell W and Z bosons, and up to 190 GeV for decays via W and Standard Model Higgs bosons.