  We present a study of cold gas absorption from a damped Lyman-$\alpha$
absorber (DLA) at redshift $z_{\rm abs}=1.946$ towards two lensed images of the
quasar J144254.78+405535.5 at redshift $z_{\rm QSO} = 2.590$. The physical
separation of the two lines of sight at the absorber redshift is $d_{\rm
abs}=0.7$~kpc based on 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 $d_{\rm abs}$. We measure column densities of HI to be
$\log N(\rm H\,\i) = 20.27\pm0.02$ and $20.34\pm0.05$ and of H$_2$ to be $\log
N(\rm H_2) = 19.7\pm0.1$ and $19.9\pm0.2$. The metallicity inferred from
sulphur is consistent with Solar metallicity for both sightlines: $[{\rm
S/H}]_A = 0.0\pm0.1$ and $[{\rm S/H}]_B = -0.1\pm0.1$. Based on the excitation
of low rotational levels of H$_2$, we constrain the temperature of the cold gas
phase to be $T=109\pm20$ and $T=89\pm25$ K for the two lines of sight. From the
relative excitation of fine-structure levels of CI, we constrain the hydrogen
volumetric densities 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 $l\approx0.1$ pc. Using the
transverse line-of-sight separation of 0.7 kpc together with the individual
cloud size, we are able to put an upper limit to the volume filling factor of
cold gas of $f_{\rm vol} < 0.2$ %. 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 (~10-30 kpc), this is consistent with the relative
incidence rate of CI absorbers and DLAs.
