A search for vector-like leptons in multilepton (two, three, or four-or-more electrons plus muons) final states with zero or more hadronic τ-lepton decays is presented. The search is performed using a dataset corresponding to an integrated luminosity of 139 fb$^{−1}$ of proton-proton collisions at a centre-of-mass energy of 13 TeV recorded by the ATLAS detector at the LHC. To maximize the separation of signal and background, a machine-learning classifier is used. No excess of events is observed beyond the Standard Model expectation. Using a doublet vector-like lepton model, vector-like leptons coupling to third-generation Standard Model leptons are excluded in the mass range from 130 GeV to 900 GeV at the 95% confidence level, while the highest excluded mass is expected to be 970 GeV.