The Intergovernmental Panel on Climate Change has projected an increased frequency of hydroclimatic extremes in its Sixth Assessment (2021). Floods are responsible for huge economic and human costs, and quantifying how the probability and magnitude of extreme flooding events are changing is key to mitigating their impacts. Spatial extreme value analysis (EVA) is a common approach, but EVA models like the max-stable process (MSP) give intractable likelihoods, making computation challenging. Composite likelihood (CL) methods provide an alternative approach, and pairwise CL which approximates the likelihood by the product of bivariate likelihoods is a leading method for inference with MSPs. Our work takes a Bayesian approach, using a deep learning model for likelihood approximation. We propose a unique computational strategy where a neural network is embedded in a density regression model to approximate the conditional distribution at one spatial location given a set of neighbors. We then use this univariate density function to approximate the joint likelihood for all locations with a Vecchia approximation. Our model is fast to fit and evaluate compared to using Gaussian Processes.