Severe thunderstorms cause substantial economic and human losses in the United States. Simultaneous high values of convective available potential energy (CAPE) and storm relative helicity (SRH) are favorable to severe weather, and both they and the composite variable (Formula presented.) can be used as indicators of severe thunderstorm activity. Their extremal spatial dependence exhibits temporal non-stationarity due to seasonality and large-scale atmospheric signals such as El Niño-Southern Oscillation (ENSO). In order to investigate this, we introduce a space-time model based on a max-stable, Brown–Resnick, field whose range depends on ENSO and on time through a tensor product spline. We also propose a max-stability test based on empirical likelihood and the bootstrap. The marginal and dependence parameters must be estimated separately owing to the complexity of the model, and we develop a bootstrap-based model selection criterion that accounts for the marginal uncertainty when choosing the dependence model. In the case study, the out-sample performance of our model is good. We find that extremes of PROD, CAPE, and SRH are generally more localized in summer and, in some regions, less localized during El Niño and La Niña events, and give meteorological interpretations of these phenomena. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.