  A gauged $U(1)_X$ symmetry appended to the Standard Model (SM) is
particularly well-motivated since it can account for the light neutrino masses
by the seesaw mechanism, explain the origin of baryon asymmetry of the universe
via leptogenesis, and help implement successful cosmological inflation with the
$U(1)_X$ breaking Higgs field as the inflaton. In this framework, we propose a
light dark matter (DM) scenario in which the $U(1)_X$ gauge boson $Z^\prime$
behaves as a DM particle in the universe. We discuss how this scenario with
$Z^\prime$ mass of a few keV and a $U(1)_X$ gauge coupling $g_X \simeq
10^{-16}$ can nicely fit the excess in the electronic recoil energy spectrum
recently reported by the XENON1T collaboration. In order to reproduce the
observed DM relic density in the presence of such a tiny gauge coupling, we
propose an extension of the model to a two-component DM scenario. The
$Z^\prime$ DM density can be comparable to the observed DM density by the
freeze-in mechanism through the coupling of $Z^\prime$ boson to a partner
Higgs-portal scalar DM with a large $U(1)_X$ charge.
