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 G0 = 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 (~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 30 115 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 centre (or ~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 findings 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.