We investigate the absorption features associated with a gas-rich dwarf galaxy using cosmological hydrodynamics simulations. Our goal is to explore whether the progenitors of the lowest mass dwarf galaxies known to harbor neutral hydrogen today (M_star~10^6 solar mass, M_halo=4x10^9 solar mass) could possibly be detected as Damped Lyman-alpha Absorbers (DLAs) over cosmic time. We trace the evolution of a single dwarf galaxy, pre-selected to contain DLAs, from the era of the first metal-free, so-called Population~III (Pop~III), stars, down to z=0, thus allowing us to study the metal enrichment history of DLAs associated with the simulated galaxy. We find that the progenitors of the simulated dwarf are expected to be seen for most of their evolution as DLAs that are contaminated by normal, Population~II, stars. The time period during which DLAs are only metal-enriched by Pop~III stars, on the other hand, is likely very brief, confined to high redshifts, z~6. The susceptibility of the dwarfs to the external UV radiation background allows them to preserve neutral gas only at the centre (a few ~100 pc). This results in a small probability that the simulated dwarf would be observed as a DLA. This study suggests that DLAs are unlikely to be hosted in the lowest mass dwarfs that can harbor neutral gas (M_halo~ 4x10^9 solar mass), below which neutral gas is unlikely to exist. However, this study does illustrate that, when detected, absorption lines provide a powerful method for probing ISM conditions inside the smallest dwarf galaxies at intermediate to high redshifts.
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