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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.