We present a study of cold gas absorption from a damped Lyman-alpha absorber (DLA) at redshift z(abs) = 1.946 toward two lensed images of the quasar J144254.78+405535.5 at redshift z(QSO) = 2.590. The physical separation of the two lines of sight at the absorber redshift is d(abs) = 0.7 kpc according to our lens model. We observe absorption lines from neutral carbon and H-2 along both lines of sight, indicating that cold gas is present on scales larger than d(abs). We measure the column densities of H I to be log N(H I) = 20.27 +/- 0.02 and 20.34 +/- 0.05 and those of H-2 to be log N(H-2) = 19.7 +/- 0.1 and 19.9 +/- 0.2. The metallicity inferred from sulphur is consistent with solar metallicity for both sightlines: S/H = 0.0 +/- 0.1 and S/H = -0.1 +/- 0.1. Based on the excitation of low rotational levels of H-2, we constrain the temperature of the cold gas phase to be T = 109 +/- 20 and T = 89 +/- 25 K for the two lines of sight. From the relative excitation of fine-structure levels of C I, we constrain the hydrogen volumetric densities to lie in the range of 40-110 cm(-3). Based on the ratio of observed column density and volumetric density, we infer the average individual "cloud" size along the line of sight to be l approximate to 0.1 pc. Using the transverse line-of-sight separation of 0.7 kpc together with the individual cloud size, we are able to place an upper limit to the volume filling factor of cold gas of f(vol) < 0.1%. Nonetheless, the projected covering fraction of cold gas must be large (close to unity) over scales of a few kpc in order to explain the presence of cold gas in both lines of sight. Compared to the typical extent of DLAs (similar to 10-30 kpc), this is consistent with the relative incidence rate of C I absorbers and DLAs.