The high-energy environment and atmospheric escape of the mini-Neptune K2-18 b

K2-18 b is a transiting mini-Neptune that orbits a nearby (38 pc) cool M3 dwarf and is located inside its region of temperate irradiation. We report on the search for hydrogen escape from the atmosphere K2-18 b using Lyman-$\alpha$ transit spectroscopy with the Space Telescope Imaging Spectrograph (STIS) instrument installed on the Hubble Space Telescope (HST). We analyzed the time-series of the fluxes of the stellar Lyman-$\alpha$ emission of K2-18 in both its blue- and redshifted wings. We found that the average blueshifted emission of K2-18 decreases by $67\mathrm{\%}\pm 18\mathrm{\%}$ during the transit of the planet compared to the pre-transit emission, tentatively indicating the presence of H atoms escaping vigorously and being blown away by radiation pressure. This interpretation is not definitive because it relies on one partial transit. Based on the reconstructed Lyman-$\alpha$ emission of K2-18, we estimate an EUV irradiation between ${10}^1-{10}^2$ erg s${}^{-1}$ cm${}^{-2}$ and a total escape rate in the order of ${10}^8$ g s${}^{-1}$. The inferred escape rate suggests that the planet will lose only a small fraction (< 1%) of its mass and retain its volatile-rich atmosphere during its lifetime. More observations are needed to rule out stellar variability effects, confirm the in-transit absorption and better assess the atmospheric escape and high-energy environment of K2-18 b.