  The most metal-deficient stars hold important clues about the early build-up
and chemical evolution of the Milky Way, and carbon-enhanced metal-poor (CEMP)
stars are of special interest. However, little is known about CEMP stars in the
Galactic bulge. In this paper, we use the large spectroscopic sample of
metal-poor stars from the Pristine Inner Galaxy Survey (PIGS) to identify CEMP
stars ([C/Fe] > +0.7) in the bulge region and to derive a CEMP fraction. We
identify 96 new CEMP stars in the inner Galaxy, of which 62 are very metal-poor
([Fe/H] < -2.0); this is more than a ten-fold increase compared to the seven
previously known bulge CEMP stars. The cumulative fraction of CEMP stars in
PIGS is $42^{\,+14\,}_{\,-13} \%$ for stars with [Fe/H] < -3.0, and decreases
to $16^{\,+3\,}_{\,-3} \%$ for [Fe/H] < -2.5 and $5.7^{\,+0.6\,}_{\,-0.5} \%$
for [Fe/H] < -2.0. The PIGS inner Galaxy CEMP fraction for [Fe/H] < -3.0 is
consistent with the halo fraction found in the literature, but at higher
metallicities the PIGS fraction is substantially lower. While this can partly
be attributed to a photometric selection bias, such bias is unlikely to fully
explain the low CEMP fraction at higher metallicities. Considering the typical
carbon excesses and metallicity ranges for halo CEMP-s and CEMP-no stars, our
results point to a possible deficiency of both CEMP-s and CEMP-no stars
(especially the more metal-rich) in the inner Galaxy. The former is potentially
related to a difference in the binary fraction, whereas the latter may be the
result of a fast chemical enrichment in the early building blocks of the inner
Galaxy.
