We demonstrate that, in contrast to most two-dimensional materials, ultrathin flakes of InSe are stable under ambient conditions. Despite their ambient stability, InSe-based nanodevices show an environmental p-type doping, suppressed by capping InSe with hexagonal boron nitride. By means of transport experiments, density functional theory and vibrational spectroscopy, we attribute the p-type doping assumed by uncapped InSe under an ambient atmosphere to the decomposition of water at Se vacancies. We have estimated the site-dependent adsorption energy of O2, N2, H2O, CO and CO2 on InSe. A stable adsorption is found only for the case of H2O, with a charge transfer of only 0.01 electrons per water molecule.