Few spatio-temporal models allow temporal non-stationarity. When modeling environmental data recorded over the last decades of the 20th century until now, it seems not reasonable to assume temporal stationarity, since it would not capture climate change effects. In this paper, we propose a space-time max-stable model for modeling some temporal non-stationarity of the spatial extremal dependence. Our model consists of a mixture of max-stable spatial processes, with a rate of mixing depending on time. We use maximum composite likelihood for estimation, model selection, and a non-stationarity test. The assessment of its performance is done through wide simulation experiments. The proposed model is used to investigate how the rainfall in the south of France evolves with time. The results demonstrate that the spatial extremal dependence is significantly non-stationary over time, with a decrease in the strength of dependence.