A distributionally robust Bayesian nonparametric regression estimator is the solution of a min-max game in which the statistician chooses a regression function of observations (i.e. an element in L2) and the adversary, knowing the statistician's selection, maximizes the mean-squared error incurred over a Wasserstein-type-2 ball around a full nonparametric Bayesian model, which we assume to be Gaussian on a suitable Hilbert space. We study this doubly infinite-dimensional game, show the existence of a Nash equilibrium and its evaluation.