All carbon materials, e.g., amorphous carbon (a-C) coatings and C60 fullerene thin films, play an important role in short-wavelength free-electron laser (FEL) research motivated by FEL optics development and prospective nanotechnology applications. Responses of a-C and C60 layers to the extreme ultraviolet (SPring-8 Compact SASE Source in Japan) and soft x-ray (free-electron laser in Hamburg) free-electron laser radiation are investigated by Raman spectroscopy, differential interference contrast, and atomic force microscopy. A remarkable difference in the behavior of covalent (a-C) and molecular (C60) carbonaceous solids is demonstrated under these irradiation conditions. Low thresholds for ablation of a fullerene crystal (estimated to be around 0.15 eV/atom for C60 vs 0.9 eV/atom for a-C in terms of the absorbed dose) are caused by a low cohesive energy of fullerene crystals. An efficient mechanism of the removal of intact C60 molecules from the irradiated crystal due to Coulomb repulsion of fullerene-cage cation radicals formed by the ionizing radiation is revealed by a detailed modeling.