The sightline to the brighter member of the gravitationally lensed quasar pair UM 673A,B intersects a damped Lyman-alpha system (DLA) at z = 1.62650 which, because of its low redshift, has not been recognised before. Our high quality echelle spectra of the pair, obtained with HIRES on the Keck I telescope, show a drop in neutral hydrogen column density N(H I) by a factor of at least 400 between UM 673A and B, indicating that the DLAs extent in this direction is much less than the 2.7 kpc separation between the two sightlines at z = 1.62650. By reassessing this new case together with published data on other quasar pairs, we conclude that the typical size (radius) of DLAs at these redshifts is R ~ (5 +/- 3) kpc, smaller than previously realised. Highly ionized gas associated with the DLA is more extended, as we find only small differences in the C IV absorption profiles between the two sightlines. Coincident with UM 673B, we detect a weak and narrow Ly-alpha emission line which we attribute to star formation activity at a rate SFR >~ 0.2 M_solar/yr. The DLA in UM 673A is metal-poor, with an overall metallicity Z_DLA ~ 1/30 Z_solar, and has a very low internal velocity dispersion. It exhibits some apparent peculiarities in its detailed chemical composition, with the elements Ti, Ni, and Zn being deficient relative to Fe by factors of 2-3. The [Zn/Fe] ratio is lower than those measured in any other DLA or Galactic halo star, presumably reflecting somewhat unusual previous enrichment by stellar nucleosynthesis. We discuss the implications of these results for the nature of the galaxy hosting the DLA.