  We identify two ultra-cool ($T_\mathrm{eff} < 4000$ K) metal-polluted (DZ)
white dwarfs WDJ2147$-$4035 and WDJ1922$+$0233 as the coolest and second
coolest DZ stars known to date with $T_\mathrm{eff} \approx 3050$ K and
$T_\mathrm{eff} \approx 3340$ K, respectively. Strong atmospheric
collision-induced absorption (CIA) causes the suppression of red optical and
infra-red flux in WDJ1922$+$0233, resulting in an unusually blue colour given
its low temperature. WDJ2147$-$4035 has moderate infra-red CIA yet has the
reddest optical colours known for a DZ white dwarf. Microphysics improvements
to the non-ideal effects and CIA opacities in our model atmosphere code yields
reasonable solutions to observations of these ultra-cool stars. WDJ2147$-$4035
has a cooling age of over 10 Gyr which is the largest known for a DZ white
dwarf, whereas WDJ1922$+$0233 is slightly younger with a cooling age of 9 Gyr.
Galactic kinematics calculations from precise Gaia EDR3 astrometry reveal these
ultra-cool DZ stars as likely members of the Galactic disc thus they could be
pivotal objects in future studies constraining an upper age limit for the disc
of the Milky Way. We present intermediate-resolution spectroscopy for both
objects, which provides the first spectroscopic observations of WDJ2147$-$4035.
Detections of sodium and potassium are made in both white dwarfs, in addition
to calcium in WDJ1922$+$0233 and lithium in WDJ2147$-$4035. We identify the
magnetic nature of WDJ2147$-$4035 from Zeeman splitting in the lithium line and
also make a tentative detection of carbon, so we classify this star as DZQH.
WDJ1922$+$0233 likely accreted planetary crust debris, while the debris
composition that polluted WDJ2147$-$4035 remains unconstrained.
