A search for supersymmetry in events with four or more charged leptons (electrons, muons and $\tau$-leptons) is presented. The analysis uses a data sample corresponding to $139fb\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of proton-proton collisions delivered by the Large Hadron Collider at $s=13$ TeV and recorded by the ATLAS detector. Four-lepton signal regions with up to two hadronically decaying $\tau$-leptons are designed to target several supersymmetric models, while a general five-lepton signal region targets any new physics phenomena leading to a final state with five charged leptons. Data yields are consistent with Standard Model expectations and results are used to set upper limits on contributions from processes beyond the Standard Model. Exclusion limits are set at the 95% confidence level in simplified models of general gauge-mediated supersymmetry, excluding higgsino masses up to $540$ GeV. In $R$-parity-violating simplified models with decays of the lightest supersymmetric particle to charged leptons, lower limits of $1.6$ TeV, $1.2$ TeV, and $2.5$ TeV are placed on wino, slepton and gluino masses, respectively.