A search for the Higgs boson decaying into a photon and a pair of electrons or muons with an invariant mass m_ℓℓ<30 GeV is presented. The analysis is performed using 139 fb⁻¹ of proton–proton collision data, produced by the LHC at a centre-of-mass energy of 13 TeV and collected by the ATLAS experiment. Evidence for the H→ℓℓγ process is found with a significance of 3.2 over the background-only hypothesis, compared to an expected significance of 2.1 for the Standard Model prediction. The best-fit value of the signal-strength parameter, defined as the ratio of the observed signal yield to the one expected in the Standard Model, is μ=1.5±0.5. The Higgs boson production cross-section times the H→ℓℓγ branching ratio for m_ℓℓ<30 GeV is determined to be 8.7_−2.7 ^+2.8 fb.