The circumgalactic medium (CGM), i.e. the gaseous haloes around galaxies, is both the reservoir of gas that fuels galaxy growth and the repository of gas expelled by galactic winds. Most cosmological, hydrodynamical simulations focus their computational effort on the galaxies themselves and treat the CGM more coarsely, which means small-scale structure cannot be resolved. We get around this issue by running zoom-in simulations of a Milky Way-mass galaxy with standard mass refinement and additional uniform spatial refinement within the virial radius. This results in a detailed view of its gaseous halo at unprecedented (1 kpc) uniform resolution with only a moderate increase in computational time. The improved spatial resolution does not impact the central galaxy or the average density of the CGM. However, it drastically changes the radial profile of the neutral hydrogen column density, which is enhanced at galactocentric radii larger than 40 kpc. The covering fraction of Lyman-Limit Systems within 150 kpc is almost doubled. We therefore conclude that some of the observational properties of the CGM are strongly resolution dependent. Increasing the resolution in the CGM, without increasing the resolution of the galaxies, is a promising and computationally efficient method to push the boundaries of state-of-the-art simulations.