One of the major conundrums in contemporary extragalactic astrophysics is the apparent overabundance of a remarkable population of UV-bright galaxies at redshifts z ≳ 9. We analysed galaxies spectroscopically observed by JWST/NIRSpec PRISM and confirmed to lie at z > 9, with a sufficient signal-to-noise to carefully model their rest-frame UV-to-optical continua and line emission. In particular, we modelled the damped Lyman-α (Lyα) absorption (DLA) features of each galaxy to place observational constraints on the gas assembly of neutral atomic hydrogen (H I) onto the galaxy halos at the onset of cosmic reionisation. Based on the derived H I column densities and star formation rate (SFR) surface densities, we show that all galaxies are highly efficient at forming stars on rapid ∼10 − 100 Myr depletion timescales, greatly in excess compared to the canonical local Universe Kennicutt-Schmidt (KS) relation and predictions from state-of-the-art galaxy formation simulations. The dense H I gas appears to also drive the offset from the fundamental-metallicity relation (FMR) of these galaxies though its dust-to-gas (DTG) ratio is seemingly consistent with values derived for local galaxies except for the lowest metallicity sight lines. Our results provide the first robust observational constraints on the impact of pristine H I gas on early galaxy assembly, and they imply that a combination of highly efficient star formation and low dust obscuration can likely explain the UV brightness of galaxies at cosmic dawn.