Our Galaxy is known to contain a central boxy/peanut-shaped bulge, yet the importance of a classical, pressure-supported component within the central part of the Milky Way is still being debated. It should be most visible at low metallicity, a regime that has not yet been studied in detail. Using metallicity-sensitive narrow-band photometry, the Pristine Inner Galaxy Survey (PIGS) has collected a large sample of metal-poor ([Fe/H] < -1.0) stars in the inner Galaxy to address this open question. We use PIGS to trace the metal-poor inner Galaxy kinematics as function of metallicity for the first time. We find that the rotational signal decreases with decreasing [Fe/H], until it becomes negligible for the most metal-poor stars. Additionally, the velocity dispersion increases with decreasing metallicity for -3.0 < [Fe/II] < -0.5, with a gradient of -44 +/- 41un s(-1)dex(-1). These observations may signal a transition between Galactic components of different metallicities and kinematics, a different mapping on to the boxy/peanut-shaped bulge for former disc stars of different metallicities and/or the secular dynamical and gravitational influence of the bar on the pressure-supported component. Our results provide strong constraints on models that attempt to explain the properties of the inner Galaxy.