Mixed models incorporating spatially correlated random effects are often used for the analysis of areal data. In this setting, spatial smoothing is introduced at the second stage of a hierarchical framework, and this smoothing is often based on a latent Gaussian Markov random field. The Markov random field provides a computationally convenient framework for modeling spatial dependence; however, the Gaussian assumption underlying commonly used models can be overly restrictive in some applications. This can be a problem in the presence of outliers or discontinuities in the underlying spatial surface, and in such settings, models based on non-Gaussian spatial random effects are useful. Motivated by a study examining geographic variation in the treatment of acute coronary syndrome, we develop a robust model for smoothing small-area health service utilization rates. The model incorporates non-Gaussian spatial random effects, and we develop a formulation for skew-elliptical areal spatial models. We generalize the Gaussian conditional autoregressive model to the non-Gaussian case, allowing for asymmetric skew-elliptical marginal distributions having flexible tail behavior. The resulting new models are flexible, computationally manageable, and can be implemented in the standard Bayesian software WinBUGS. We demonstrate performance of the proposed methods and comparisons with other commonly used Gaussian and non-Gaussian spatial prior formulations through simulation and analysis in our motivating application, mapping rates of revascularization for patients diagnosed with acute coronary syndrome in Quebec, Canada.