We propose a new statistical model that captures the conditional dependence among extreme events in a spatial domain. This model may for instance be used to describe catastrophic events such as earthquakes, floods, or hurricanes in certain regions, and in particular to predict extreme values at unmonitored sites. The proposed model is derived as follows. The block maxima at each location are assumed to follow a Generalized Extreme Value (GEV) distribution. Spatial dependence is modeled in two complementary ways. The GEV parameters are coupled through a thin-membrane model, a specific type of Gaussian graphical model often used as smoothness prior. The extreme events, on the other hand, are coupled through a copula Gaussian graphical model with the precision matrix corresponding to a (generalized) thin-membrane model. We then derive inference and interpolation algorithms for the proposed model. The approach is validated on synthetic data as well as real data related to hurricanes in the Gulf of Mexico. Numerical results suggest that it can accurately describe extreme events in spatial domain, and can reliably interpolate extreme values at arbitrary sites.