  We consider a local $U(1)_{B-L}$ extension of Zee-Babu model to explain the
recently observed 3.5 keV X-ray line signal. The model has three Standard model
(SM)-singlet Dirac fermions with different $U(1)_{B-L}$ charges. A complex
scalar field charged under $U(1)_{B-L}$ is introduced to break the $U(1)_{B-L}$
symmetry. After $U(1)_{B-L}$ symmetry breaking a remnant discrete symmetry
stabilizes the lightest state of the Dirac fermions, which can be a stable dark
matter (DM). The second lightest state, if mass splitting with the stable DM is
about 3.5 keV, decays dominantly to the stable DM and 3.5 keV photon through
two-loop diagrams, explaining the X-ray line signal. Two-loop suppression of
the decay amplitude makes its lifetime much longer than the age of the universe
and it can be a decaying DM candidate in large parameter region. We also
introduce a real scalar field which is singlet under both the SM and
$U(1)_{B-L}$ and can explain the current relic abundance of the Dirac fermionic
DMs. If the mixing with the SM Higgs boson is small, it does not contribute to
DM direct detection. The main contribution to the scattering of DM off atomic
nuclei comes from the exchange of $U(1)_{B-L}$ gauge boson, $Z'$, and is
suppressed below current experimental bound when $Z'$ mass is heavy ($\gtrsim
10$ TeV). If the singlet scalar mass is about 0.1-10 MeV, the DM
self-interaction can be large enough to solve small scale structure problems in
simulations with the cold DM, such as, the core-vs-cusp problem and
too-big-to-fail problem.
