The use of graphene electronics in space will depend on the radiation hardness of graphene. The damage threshold of graphene samples, subjected to 2 MeV H+ irradiation, was found to increase with layer number and also when the graphene layer was supported by a substrate. The thermal properties of graphene as a function of the number of layers or as influenced by the substrate argue against a thermal model for the production of damage by the ion beam. We propose a model of intense electronically-stimulated surface desorption of the atoms as the most likely process for this damage mechanism. © 2010 Elsevier Ltd. All rights reserved.