Gaussian process regression is a classical technique for (Bayesian) non-parametric regression estimation. We propose and study a distributionally robust version of Gaussian process regression based on a min-max game between the statistician and an adversary that introduces non-parametric perturbations of the prior based on an optimal transport cost. The perturbations are allowed both to be outside the support of the Gaussian prior and have arbitrary distributions. We show that this game has a unique Nash equilibrium and provide computational algorithms for computing the associated distributionally robust regression function.