  A search for events with a dark photon produced in association with a dark
Higgs boson via rare decays of the Standard Model $Z$ boson is presented, using
139 fb$^{-1}$ of $\sqrt{s} = 13$ TeV proton-proton collision data recorded by
the ATLAS detector at the Large Hadron Collider. The dark Higgs boson decays
into a pair of dark photons, and at least two of the three dark photons must
each decay into a pair of electrons or muons, resulting in at least two
same-flavor opposite-charge lepton pairs in the final state. The data are found
to be consistent with the background prediction, and upper limits are set on
the dark photon's coupling to the dark Higgs boson times the kinetic mixing
between the Standard Model photon and the dark photon,
$\alpha_{D}\varepsilon^2$, in the dark photon mass range of $[5, 40]$ GeV
except for the $\Upsilon$ mass window $[8.8, 11.1]$ GeV. This search explores
new parameter space not previously excluded by other experiments.
