We present global VLBI observations of the 21-cm transition of atomic hydrogen seen in absorption against the radio source J0855+5751. The foreground absorber (SDSS J085519.05+575140.7) is a dwarf galaxy at z = 0.026. As the background source is heavily resolved by VLBI, the data allow us to map the properties of the foreground H i gas with a spatial resolution of 2 pc. The absorbing gas corresponds to a single coherent structure with an extent >35 pc, but we also detect significant and coherent variations, including a change in the H i optical depth by a factor of 5 across a distance of ≲ 6 pc. The large size of the structure provides support for the Heiles & Troland model of the interstellar medium, as well as its applicability to external galaxies. The large variations in H i optical depth also suggest that caution should be applied when interpreting TS measurements from radio-detected DLAs. In addition, the distorted appearance of the background radio source is indicative of a strong jet–cloud interaction in its host galaxy. We have measured its redshift (z = 0.541 86) using optical spectroscopy on the William Herschel Telescope and this confirms that J0855+5751 is an FR II radio source with a physical extent of <1 kpc and supports the previous identification of this source as a compact symmetric object. These sources often show absorption associated with the host galaxy and we suggest that both H i and OH should be searched for in J0855+5751.