Aims. Cosmic-ray-induced sputtering is one of the important desorption mechanisms at work in astrophysical environments. The chemical evolution observed in high-density regions, from dense clouds to protoplanetary disks, and the release of species condensed on dust grains, is one key parameter to be taken into account in interpretations of both observations and models. Methods. This study is part of an ongoing systematic experimental determination of the parameters to consider in astrophysical cosmic ray sputtering. As was already done for water ice, we investigated the sputtering yield as a function of ice mantle thickness for the two next most abundant species of ice mantles, carbon monoxide and carbon dioxide, which were exposed to sever...