We introduce a method for performing a robust Bayesian analysis of non-Gaussianity present in pulsar timing data, simultaneously with the pulsar timing model, and additional stochastic parameters such as those describing red spin noise and dispersion measure variations. The parameters used to define the presence of non-Gaussianity are zero for Gaussian processes, giving a simple method of defining the strength of non-Gaussian behaviour. We use simulations to show that assuming Gaussian statistics when the noise in the data is drawn from a non-Gaussian distribution can significantly increase the uncertainties associated with the pulsar timing model parameters. We then apply the method to the publicly available 15 year Parkes Pulsar Timing Array data release 1 dataset for the binary pulsar J0437$-$4715. In this analysis we present a significant detection of non-Gaussianity in the uncorrelated non-thermal noise, but we find that it does not yet impact the timing model or stochastic parameter estimates significantly compared to analysis performed assuming Gaussian statistics. The methods presented are, however, shown to be of immediate practical use for current European Pulsar Timing Array (EPTA) and International Pulsar Timing Array (IPTA) datasets.
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