Accurate spatiotemporal modeling of conditions leading to moderate and large wildfires provides better understanding of mechanisms driving fire-prone ecosystems and improves risk management. Here, we develop a joint model for the occurrence intensity and the wildfire size distribution, by combining extreme-value theory and point processes within a novel Bayesian hierarchical model, and use it to study daily summer wildfire data for the French Mediterranean basin during 1995–2018. The occurrence component models wildfire ignitions as a spatiotemporal log-Gaussian Cox process. Burnt areas are numerical marks attached to points and are considered as extreme if they exceed a high threshold. The size component is a two-component mixture varying in space and time that jointly models moderate and extreme fires. We capture nonlinear influence of covariates (Fire Weather Index, forest cover) through component-specific smooth functions which may vary with season. We propose estimating shared random effects between model components to reveal and interpret common drivers of different aspects of wildfire activity. This increases parsimony and reduces estimation uncertainty, giving better predictions. Specific stratified subsampling of zero counts is implemented to cope with large observation vectors. We compare and validate models through predictive scores and visual diagnostics. Our methodology provides a holistic approach to explaining and predicting the drivers of wildfire activity and associated uncertainties.