Extreme wildfires are a significant cause of human death and biodiversity destruction within countries that encompass the Mediterranean Basin. Recent worrying trends in wildfire activity (i.e., occurrence and spread) suggest that wildfires are likely to be highly impacted by climate change. In order to facilitate appropriate risk mitigation, we must identify the main drivers of extreme wildfires and assess their spatio-temporal trends, with a view to understanding the impacts of global warming on fire activity. We analyse the monthly burnt area due to wildfires over a region encompassing most of Europe and the Mediterranean Basin from 2001 to 2020, and identify high fire activity during this period in Algeria, Italy and Portugal. We build an extreme quantile regression model with a high-dimensional predictor set describing meteorological conditions, land cover usage, and orography. To model the complex relationships between the predictor variables and wildfires, we use a hybrid statistical deep-learning framework that can disentangle the effects of vapour-pressure deficit (VPD), air temperature, and drought on wildfire activity. Our results highlight that whilst VPD, air temperature, and drought significantly affect wildfire occurrence, only VPD affects wildfire spread. To gain insights into the effect of climate trends on wildfires in the near future, we focus on August 2001 and perturb temperature according to its observed trends (median over Europe: +0.04K per year). We find that, on average over Europe, these trends lead to a relative increase of 17.1\% and 1.6\% in the expected frequency and severity, respectively, of wildfires in August 2001, with spatially non-uniform changes in both aspects.