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 $$\sqrt{s} = 13$$ $\sqrt{s}=13$ TeV pp collisions recorded with the ATLAS detector at the LHC, corresponding to an integrated luminosity of 139 $$\hbox {fb}^{-1}$$ ${\text{fb}}^{-1}$ . 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 ( $${\tilde{\chi }}^\pm _1$$ ${\tilde{\chi }}_1^{\pm }$ ) and neutralinos ( $${\tilde{\chi }}^0_2$$ ${\tilde{\chi }}_2^0$ ) are considered. For pure higgsino $${\tilde{\chi }}^\pm _1{\tilde{\chi }}^0_2$$ ${\tilde{\chi }}_1^{\pm }{\tilde{\chi }}_2^0$ pair-production scenarios, exclusion limits at 95% confidence level are set on $${\tilde{\chi }}^0_2$$ ${\tilde{\chi }}_2^0$ masses up to 210 GeV. Limits are also set for pure wino $${\tilde{\chi }}^\pm _1{\tilde{\chi }}^0_2$$ ${\tilde{\chi }}_1^{\pm }{\tilde{\chi }}_2^0$ pair production, on $${\tilde{\chi }}^0_2$$ ${\tilde{\chi }}_2^0$ 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.