Current dependence models for spatial extremes are based upon max-stable processes. Within this class, there are few inferentially viable models available, and we propose one further model. More problematic are the restrictive assumptions that must be made when using max-stable processes to model dependence for spatial extremes: it must be assumed that the dependence structure of the observed extremes is compatible with a limiting model that holds for all events more extreme than those that have already occurred. This problem has long been acknowledged in the context of finite-dimensional multivariate extremes, in particular when data display dependence at observable levels, but are independent in the limit. We propose a flexible class of models that is suitable for such data in a spatial context. In addition, we consider the situation where the extremal dependence structure may vary with distance. We apply our models to spatially referenced significant wave height data from the North Sea, finding evidence that their extremal structure is not compatible with a limiting dependence model.