In this paper, we present an extension of the class of uncertain-input models to handle cases of measurements with outliers. The general uncertain-input model framework allows us to treat system identification problems in which a linear system, represented by its impulse response, is subject to an input about which we have partial information. Both the impulse response and the input are modeled ... [Show full abstract] as Gaussian processes and the kernels are used to encode the information available. The whole model is then estimated using an approximate empirical Bayes approach. We extend the uncertain-input model framework to non-Gaussian measurement models by considering the noise precisions as realizations of a Gamma prior. We validate the approach on a dataset of linear systems and on a dataset of Hammerstein systems where the measurements are corrupted by outliers.