Search for gas from the disintegrating rocky exoplanet K2-22b

[Abridged] Aims. We searched for circumplanetary sodium and ionized calcium gas around the disintegrating rocky exoplanet K2-22 b to constrain its gas-loss and sublimation processes. Methods. We observed four transits of K2-22 b with X-shooter on ESO's Very Large Telescope to obtain time-series of intermediate-resolution (R $\sim$ 11400) spectra. Our analysis focused on the two sodium D lines (588.995 nm and 589.592 nm) and the Ca$^{+}$ triplet (849.802 nm, 854.209 nm and 866.214 nm). Planet-related absorption is searched for in the velocity rest frame of the planet, which changes from $\pm$66 kms$^{-1}$ during the transit. Results. Since K2-22 b exhibits highly variable transit depths, we analyzed the individual nights and their average. By injecting signals we reached 5$\sigma$ upper-limits on the individual nights that ranged from 11% - 13% and 1.7% - 2.0% for the tail's sodium and ionized calcium absorption, respectively. Night 1 was contaminated by its companion star so we considered weighted averages with and without Night 1 and quote conservative 5$\sigma$ limits without Night 1 of 9% and 1.4%, respectively. Assuming their mass fractions to be similar to those in the Earth's crust, these limits correspond to scenarios in which 0.04% and 35% of the transiting dust is sublimated and observed as absorbing gas. However, this assumes the gas to be co-moving with the planet. We show that for the high irradiation environment of K2-22 b, sodium and ionized calcium could be quickly accelerated to 100s of km s$^{-1}$ due to radiation pressure and entrainment by the stellar wind, making them much more difficult to detect. No evidence for such possibly broad and blue-shifted signals are seen in our data. Conclusions. Future observations aimed at observing circumplanetary gas should take into account the possible broad and blue-shifted velocity field of atomic and ionized species.