The spatial modelling of extreme snow is important for adequate risk management in Alpine and high altitude countries. A natural approach to such modelling is through the theory of max-stable processes, an infinite-dimensional extension of multivariate extreme value theory. In this paper we describe the application of such processes in modelling the spatial dependence of extreme snow depth in Switzerland, based on data for the winters 1966--2008 at 101 stations. The models we propose rely on a climate transformation that allows us to account for the presence of climate regions and for directional effects, resulting from synoptic weather patterns. Estimation is performed through pairwise likelihood inference and the models are compared using penalized likelihood criteria. The max-stable models provide a much better fit to the joint behaviour of the extremes than do independence or full dependence models.