A search for pair production of doubly charged Higgs bosons (H-+/-+/-), 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-+/-+/- -> l(+/-)l'(+/-) where l, l' = 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-+/-+/-), at 95% confidence level. Assuming that the branching ratios to each of the possible leptonic final states are equal, B(H-+/-+/- -> e(+/-)e(+/-)) = B(H-+/-+/- -> e(+/-)mu(+/-)) = B(H-+/-+/- -> mu(+/-)mu(+/-)) = B(H-+/-+/- -> e(+/-)tau(+/-)) = B(H-+/-+/- -> mu(+/-)tau(+/-)) = B(H-+/-+/- -> tau(+/-)tau(+/-)) = 1/6, the observed (expected) lower limit on the mass of a doubly charged Higgs boson is 1080GeV(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-+/-+/-) = 900 GeV (880 GeV).