  A search for pair production of doubly charged Higgs bosons ($H^{\pm \pm}$),
each decaying into a pair of prompt, isolated, highly energetic leptons with
the same electric charge, is presented. The search uses a proton-proton
collision data sample at a centre-of-mass energy of 13 TeV corresponding to an
integrated luminosity of 139 fb$^{-1}$ recorded by the ATLAS detector during
Run 2 of the Large Hadron Collider. This analysis focuses on same-charge
leptonic decays, $H^{\pm \pm} \rightarrow \ell^{\pm} \ell^{\prime \pm}$ where
$\ell, \ell^\prime=e, \mu, \tau$, in two-, three-, and four-lepton channels,
but only considers final states which include electrons or muons. No evidence
of a signal is observed. Corresponding limits on the production cross-section
and consequently a lower limit on $m(H^{\pm \pm})$ are derived at 95%
confidence level. Assuming that the branching ratios to each of the possible
leptonic final states are equal, $\mathcal{B}(H^{\pm \pm} \rightarrow e^\pm
e^\pm) = \mathcal{B}(H^{\pm \pm} \rightarrow e^\pm \mu^\pm) =
\mathcal{B}(H^{\pm \pm} \rightarrow \mu^\pm \mu^\pm) = \mathcal{B}(H^{\pm \pm}
\rightarrow e^\pm \tau^\pm) = \mathcal{B}(H^{\pm \pm} \rightarrow \mu^\pm
\tau^\pm) = \mathcal{B}(H^{\pm \pm} \rightarrow \tau^\pm \tau^\pm) = 1/6$, the
observed lower limit on the mass of a doubly charged Higgs boson is 1080 GeV
within the left-right symmetric type-II seesaw model, which is an improvement
over previous limits. Additionally, a lower limit of $m(H^{\pm \pm})$ = 900 GeV
is obtained in the context of the Zee-Babu neutrino mass model.
