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 sqrt{s} = 13 TeV pp collisions recorded with the ATLAS detector at the LHC, corresponding to an integrated luminosity of 139 hbox {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 ({tilde{chi }}^pm _1) and neutralinos ({tilde{chi }}^0_2) are considered. For pure higgsino {tilde{chi }}^pm _1{tilde{chi }}^0_2 pair-production scenarios, exclusion limits at 95% confidence level are set on {tilde{chi }}^0_2 masses up to 210 GeV. Limits are also set for pure wino {tilde{chi }}^pm _1{tilde{chi }}^0_2 pair production, on {tilde{chi }}^0_2 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.