Extreme wildfires continue to be 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, it is imperative to 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. To this end, 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 eastern Europe, 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, for the domain. To model the complex relationships between the predictor variables and wildfires, we make use of a hybrid statistical deep-learning framework that allows us to 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 extreme wildfire spread. Furthermore, to gain insights into the effect of climate change on wildfire activity in the near future, we perturb VPD and temperature according to their observed trends and find evidence that global warming may lead to spatially non-uniform changes in wildfire activity.