AbstractIn ecological systems, extremes can happen in time, such as population crashes, or in space, such as rapid range contractions. However, current methods for joint inference about temporal and spatial dynamics (e.g., spatiotemporal modeling with Gaussian random fields) may perform poorly when underlying processes include extreme events. Here we introduce a model that allows for extremes to occur simultaneously in time and space. Our model is a Bayesian predictive‐process GLMM (generalized linear mixed‐effects model) that uses a multivariate‐t distribution to describe spatial random effects. The approach is easily implemented with our flexible R package glmmfields. First, using simulated data, we demonstrate the ability to recapture spatiotemporal extremes, and explore the consequences of fitting models that ignore such extremes. Second, we predict tree mortality from mountain pine beetle (Dendroctonus ponderosae) outbreaks in the U.S. Pacific Northwest over the last 16 yr. We show that our approach provides more accurate and precise predictions compared to traditional spatiotemporal models when extremes are present. Our R package makes these models accessible to a wide range of ecologists and scientists in other disciplines interested in fitting spatiotemporal GLMMs, with and without extremes.