  We present a comprehensive metallicity analysis of the Sagittarius dwarf
spheroidal galaxy (Sgr dSph) using $Pristine\,CaHK$ photometry. We base our
member selection on $Gaia$ EDR3 astrometry applying a magnitude limit at $G_{0}
= 17.3$, and our population study on the metallicity-sensitive photometry from
the $Pristine$ Inner Galaxy Survey (PIGS). Working with photometric
metallicities instead of spectroscopic metallicities allows us to cover an
unprecedented large area ($\sim 100$ square degrees) of the dwarf galaxy, and
to study the spatial distribution of its members as function of metallicity
with little selection effects. Our study compares the spatial distributions of
a metal-poor population of 9719 stars with [Fe/H] $< -1.3$ and a metal rich one
of 30115 stars with [Fe/H] $> -1.0$. The photometric Sgr sample also allows us
to assemble the largest sample of 1150 very metal-poor Sgr candidates ([Fe/H]
$< -2.0$). By investigating and fitting the spatial properties of the
metal-rich and metal-poor population, we find a negative metallicity gradient
which extends up to 12 degrees from the Sgr center (or $\sim 5.5$ kpc at the
distance of Sgr), the limit of our footprint. We conclude that the relative
number of metal-poor stars increases in the outer areas of the galaxy, while
the central region is dominated by metal-rich stars. These finding suggest an
outside-in formation process and are an indication of the extended formation
history of Sgr, which has been affected by the tidal interaction between Sgr
and the Milky Way.
