  From temperature and magnetic field dependent point-contact spectroscopy on
the ferromagnetic superconductor Sr$_{0.5}$Ce$_{0.5}$FBiS$_2$ (bulk
superconducting $T_c$ = 2.5 K) we observe (a) a pseudogap in the normal state
that sustains to a remarkably high temperature of 40 K and (b) two-fold
enhancement of $T_c$ upto 5 K in the point-contact geometry. In addition,
Andreev reflection spectroscopy reveals a superconducting gap of 6 meV for
certain point-contacts suggesting that the mean field $T_c$ of this system
could be approximately 40 K, the onset temperature of pseudo-gap. Our results
suggest that quantum fluctuations originating from other competing orders in
Sr$_{0.5}$Ce$_{0.5}$FBiS$_2$ forbid a global phase coherence at high
temperatures thereby suppressing $T_c$. Apart from the known ordering to a
ferromagnetic state, our first-principles calculations reveal nesting of a
multi-band Fermi surface and a significant electron-phonon coupling that could
result in charge density wave-like instabilities.
