At the low temperatures found in the interior of dense clouds and circumstellar regions, along with H$_2$O and smaller amounts of species such as CO, CO$_2$, or CH$_3$OH, the infrared features of CH$_4$ have been observed on icy dust grains. Ultraviolet (UV) photons induce different processes in ice mantles, affecting the molecular abundances detected in the gas-phase. This work aims to understand the processes that occur in a pure CH$_4$ ice mantle submitted to UV irradiation. We studied photon-induced processes for the different photoproducts arising in the ice upon UV irradiation. Experiments were carried out in ISAC, an ultra-high vacuum chamber equipped with a cryostat and an F-type UV-lamp reproducing the secondary UV-field induced by cosmic rays in dense clouds. Infrared spectroscopy and quadrupole mass spectrometry were used to monitor the solid and gas-phase, respectively, during the formation, irradiation, and warm-up of the ice. Direct photodesorption of pure CH$_4$ was not observed. UV photons form CH$_x\cdot$ and H$\cdot$ radicals, leading to photoproducts such as H$_2$, C$_2$H$_2$, C$_2$H$_6$, and C$_3$H$_8$. Evidence for the photodesorption of C$_2$H$_2$ and photochemidesorption of C$_2$H$_6$ and C$_3$H$_8$ was found, the latter species is so far the largest molecule found to photochemidesorb. $^{13}$CH$_4$ experiments were also carried out to confirm the reliability of these results.