A search for light long-lived neutral particles with masses in the O(MeV–GeV) range is presented. The analysis targets the production of long-lived dark photons in the decay of a Higgs boson produced via gluon–gluon fusion or in association with a W boson. Events that contain displaced collimated Standard Model fermions reconstructed in the calorimeter or muon spectrometer are selected in 139 fb⁻¹ of s = 13 TeV pp collision data collected by the ATLAS detector at the LHC. Background estimates for contributions from Standard Model processes and instrumental effects are extracted from data. The observed event yields are consistent with the expected background. Exclusion limits are reported on the production cross-section times branching fraction as a function of the mean proper decay length cτ of the dark photon, or as a function of the dark-photon mass and kinetic mixing parameter that quantifies the coupling between the Standard Model and potential hidden (dark) sectors. A Higgs boson branching fraction above 1% is excluded at 95% CL for a Higgs boson decaying into two dark photons for dark-photon mean proper decay lengths between 10 mm and 250 mm and dark photons with masses between 0.4 GeV and 2 GeV. [Figure not available: see fulltext.].