  The non-thermal dark matter (DM) production via the so-called freeze-in
mechanism provides a simple alternative to the standard thermal WIMP scenario.
In this work, we consider a popular $U(1)_{B-L}$ extension of the standard
model (SM) in the context of inverse seesaw mechanism which has at least one
(fermionic) FIMP DM candidate. Due to the added $\mathbb{Z}_{2}$ symmetry, a SM
gauge singlet fermion, with mass of order keV, is stable and can be a warm DM
candidate. Also, the same $\mathbb{Z}_{2}$ symmetry helps the lightest
right-handed neutrino, with mass of order GeV, to be a stable or long-lived
particle by making a corresponding Yukawa coupling very small. This provides a
possibility of a two component DM scenario as well. Firstly, in the absence of
a GeV DM component (i.e., without tuning its corresponding Yukawa coupling to
be very small), we consider only a keV DM as a single component DM, which is
produced by the freeze-in mechanism via the decay of the extra $Z'$ gauge boson
associated to $U(1)_{B-L}$ and can consistently explain the DM relic density
measurements. In contrast with most of the existing literature, we have found a
reasonable DM production from the annihilation processes. After numerically
studying the DM production, we show the dependence of the DM relic density as a
function of its relevant free parameters. We use these results to obtain the
parameter space regions that are compatible with the DM relic density bound.
Secondly, we study a two component DM scenario and emphasize that the current
DM relic density bound can be satisfied for a wide range of parameter space.
