The Pristine survey XIII: Uncovering the very metal-poor tail of the
  thin disc
  We evaluate the rotational velocity of stars observed by the Pristine survey
towards the Galactic anticentre, spanning a wide range of metallicities from
the extremely metal-poor regime ($\mathrm{[Fe/H]}<-3$) to nearly solar
metallicity. In the Galactic anticentre direction, the rotational velocity
($V_{\phi}$) is similar to the tangential velocity in the galactic longitude
direction ($V_{\ell}$). This allows us to estimate $V_{\phi}$ from Gaia early
data-release 3 (Gaia EDR3) proper motions for stars without radial velocity
measurements. This substantially increases the sample of stars in the outer
disc with estimated rotational velocities. Our stellar sample towards the
anticentre is dominated by a kinematical thin disc with a mean rotation of
$\sim -220$ km $\mathrm{s}^{-1}$. However, our analysis reveals the presence of
more stellar substructures. The most intriguing is a well populated extension
of the kinematical thin disc down to $\mathrm{[Fe/H]} \sim -2$. A scarser fast
rotating population reaching the extremely metal-poor regime, down to
$\mathrm{[Fe/H]} \sim -3.5$ is also detected, but without statistical
significance to unambiguously state whether this is the extremely metal-poor
extension of the thin disc or the high rotating tail of hotter structures (like
the thick disc or the halo). In addition, a more slowly rotating kinematical
thick disc component is also required to explain the observed $V_{\ell}$
distribution at $\mathrm{[Fe/H]} > -1.5$. Furthermore, we detect signatures of
a "heated disc", the so-called Splash, at metallicities higher than $\sim-1.5$.
Finally, at $\mathrm{[Fe/H]} < -1.5$ our anticentre sample is dominated by a
kinematical halo with a net prograde motion.
