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 $log N(rm H,i) = 20.27pm0.02$ and $20.34pm0.05$ and of H$_2$ to be $log N(rm H_2) = 19.7pm0.1$ and $19.9pm0.2$. The metallicity inferred from sulphur is consistent with Solar metallicity for both sightlines: $[{rm S/H}]_A = 0.0pm0.1$ and $[{rm S/H}]_B = -0.1pm0.1$. Based on the excitation of low rotational levels of H$_2$, we constrain the temperature of the cold gas phase to be $T=109pm20$ and $T=89pm25$ 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 $lapprox0.1$ 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.
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