  We derive structure formation limits on dark matter (DM) composed of
keV-scale axion-like particles (ALPs), produced via freeze-in through the
interactions with photons and Standard Model (SM) fermions. We employ
Lyman-alpha (Ly-{\alpha}) forest data sets as well as the observed number of
Milky Way (MW) subhalos. We compare results obtained using Maxwell-Boltzmann
and quantum statistics for describing the SM bath. It should be emphasized that
the presence of logarithmic divergences complicates the calculation of the
production rate, which can not be parameterized with a simple power law
behaviour. The obtained results, in combination with X-ray bounds, exclude the
possibility for a photophilic "frozen-in" ALP DM with mass below $\sim
19\,\mathrm{keV}$. For the photophobic ALP scenario, in which DM couples
primarily to SM fermions, the ALP DM distribution function is peaked at
somewhat lower momentum and hence for such realization we find weaker limits on
DM mass. Future facilities, such as the upcoming Vera C. Rubin observatory,
will provide measurements with which the current bounds can be significantly
improved to $\sim 80\,\mathrm{keV}$.
