  It is challenging to resolve the small-scale problem for dark matter being a
weakly-interacting massive particle. We attempt to address this issue by
proposing a self-interacting freeze-in dark matter via dark photon. In this
model, the dark matter obtains the observed relic abundance via Standard Model
$\gamma$ and $Z$ boson induced freeze-in processes, whereas the dark matter
force mediator has a negligible relic abundance and a lifetime larger than the
age of Universe. We place constraints in classical and resonant regime
resolving the small-scale problem from CMB, $X/\gamma$-ray, Supernova 1987A and
out-of-equilibrium condition. It turns out that the CMB constraint on dark
matter annihilations is satisfied despite large Sommerfeld effect taking place,
while the other constraints are trivially accommodated due to various
millicharge induced suppressions. Finally we briefly discuss future
cosmological tests on such freeze-in dark matter model.
