  Over the past few years, there has been a hint of the $\gamma$-ray excess
observed by the Fermi-LAT satellite borne telescope from the regions
surrounding the Galactic Centre at an energy range $\sim 1$-$3$ GeV. The nature
of this excess $\gamma$-ray spectrum is found to be consistent with the
$\gamma$-ray emission expected from dark matter annihilation at the Galactic
Centre while disfavouring other known astrophysical sources as the possible
origin of this phenomena. It is also reported that the spectrum and morphology
of this excess $\gamma$-rays can well be explained by the dark matter particles
having mass in the range $30\sim 40$ GeV annihilating significantly into ${\rm
b} \bar{\rm b}$ final state with an annihilation cross section $\sigma {\rm
v}\sim (1.4$ - $2.0)\times 10^{-26}$ cm$^3/$s at the Galactic Centre. In this
work, we propose a two component dark matter model where two different types of
dark matter particles namely a complex scalar and a Dirac fermion are
considered. The stability of both the dark sector particles are maintained by
virtue of an additional local U$(1)_{\rm X}$ gauge symmetry. We find that our
proposed scenario can provide a viable explanation for this anomalous excess
$\gamma$-rays besides satisfying all the existing relevant theoretical as well
as experimental and observational bounds from LHC, PLANCK and LUX
collaborations. The allowed range of "effective annihilation cross section" of
lighter dark matter particle for the ${\rm b} \bar{\rm b}$ annihilation channel
thus obtained, is finally compared with the limits reported by the Fermi-LAT
and DES collaborations using data from various dwarf spheroidal galaxies.
