We reveal multiple components of an interacting galaxy system at $z\approx3.35$ through a detailed analysis of the exquisite high-resolution Keck/HIRES spectrum of the afterglow of a gamma-ray burst (GRB). Through Voigt-profile fitting of absorption lines from the Lyman-series, we constrain the neutral hydrogen column density to $N_{\mathrm{HI}} \leq 10^{18.35}$ cm$^{-2}$ for the densest of four distinct systems at the host redshift of GRB~080810, among the lowest $N_{\mathrm{HI}}$ ever observed in a GRB host, despite the line of sight passing within a projected 5 kpc of the galaxy centres. By detailed analysis of the corresponding metal absorption lines, we derive chemical, ionic and kinematic properties of the individual absorbing systems, and thus build a picture of the host as a whole. Striking differences between the systems imply that the line of sight passes through several phases of gas: the star-forming regions of the GRB host; enriched material in the form of a galactic outflow; the hot and ionised halo of a second, interacting galaxy falling towards the host at a line-of-sight velocity of 700 km s$^{-1}$; and a cool, metal-poor cloud which may represent one of the best candidates yet for the inflow of metal-poor gas from the intergalactic medium.

18 pages, 15 figures. A&A, Accepted. Updated to accepted version