A search for pair production of doubly charged Higgs bosons ($H^{\pm \pm }$), each decaying into a pair of prompt, isolated, and 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 (LHC). 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 upper limits on the production cross-section of a doubly charged Higgs boson are derived, as a function of its mass $m(H^{\pm \pm })$, 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 (expected) lower limit on the mass of a doubly charged Higgs boson is 1080 GeV (1065 GeV) within the left-right symmetric type-II seesaw model, which is the strongest limit to date produced by the ATLAS Collaboration. Additionally, this paper provides the first direct test of the Zee–Babu neutrino mass model at the LHC, yielding an observed (expected) lower limit of $m(H^{\pm \pm })$ = 900 GeV (880 GeV).