  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 $139\,\mbox{fb\(^{-1}\)}$ of proton-proton collisions
delivered by the Large Hadron Collider at $\sqrt{s}=13$ 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.
