  In earlier work we showed that a dark matter halo with a virial mass of
$10^7$ M$_\odot$ can survive feedback from its own massive stars and form stars
for $\gtrsim100$ Myr. We also found that our modelled systems were consistent
with observations of ultrafaint dwarfs (UFDs), the least massive known
galaxies. Very metal-poor damped Lyman-$\alpha$ systems (DLAs) recently
identified at $z\sim2$ may represent the gas that formed at least some of the
observed stars in UFDs. We compare projected sightlines from our simulations to
the observed metal-poor DLAs and find that our models can reach the densities
of the observed sightlines; however the metallicities are inconsistent with the
single supernova simulations, suggesting enrichment by multiple supernovae. We
model two scenarios for the history of these systems. The first explains the
gas abundances in DLAs by a single burst of star formation. This model can
produce the observed DLA abundances, but does not provide an explanation as to
why the DLAs show suppressed [$\alpha$/Fe] compared to the stellar population
of UFDs. The second scenario splits the DLAs into a population which is
enriched by a single burst, and a population that is enriched by a second burst
after the accretion of metal-poor gas. In this scenario, the suppressed average
[$\alpha$/Fe] in DLAs compared to UFDs results from enrichment of second-burst
systems by Type Ia supernovae.
