Crystalline films of pentacene molecules, two to four monolayers in thickness, are grown via in situ sublimation on silicon substrates in the ultrahigh vacuum chamber of a low-energy electron microscope. It is observed that the diffraction pattern of the pentacene layers fades upon irradiation with low-energy electrons. The damage cross section is found to increase by more than an order of magnitude for electron energies from 0 to 10 eV and by another order of magnitude from 10 to 40 eV. Close to 0 eV, damage is virtually nil. Creation of chemically reactive atomic centers after electron attachment or impact ionization is thought to trigger chemical reactions between neighboring molecules that gradually transform the layer into a disordered carbon nanomembrane. Additionally, diminishing spectroscopic features related to the unoccupied band structure of the layers, accompanied by loss of definition in real-space images, and an increase in the background intensity of diffraction images during irradiation point to chemical changes and formation of a disordered layer.