Traditional spatial models such as Gaussian processes are poorly suited for modeling extreme events. Spatial extreme value models are theoretically justified but give intractable likelihoods, making computation challenging for small datasets and prohibitive for continental-scale studies. We propose a process mixture model which specifies spatial dependence in extreme values as a convex combination of a Gaussian process and a max-stable process, yielding desirable tail dependence properties but intractable likelihoods. To address this, we employ a unique computational strategy where a feed-forward neural network is embedded in a density regression model to approximate the conditional distribution at one spatial location given a set of neighbors. The joint likelihood for all locations is then expressed in terms of the univariate densities by way of a Vecchia approximation. As a demonstrative example, the process mixture model is used to analyze changes in annual maximum streamflow within the US over the last 50 years, and is able to detect areas which show increases in extreme streamflow over time.