Context. Methanol is a common component of interstellar and circumstellar ice mantles and is often used as an evolution indicator in star-forming regions. The observations of gas-phase methanol in the interiors of dense molecular clouds at temperatures as low as 10 K suggest that non-thermal ice desorption must be active. Ice photodesorption has been proposed to explain the abundances of gas-phase molecules toward the coldest regions. Aims. Laboratory experiments were performed to investigate the potential photodesorption of methanol toward the coldest regions. Methods. Solid methanol was deposited at 8 K and UV-irradiated at various temperatures starting from 8 K. The irradiation of the ice was monitored by means of infrared spectroscopy and the molecules in the gas phase were detected using quadrupole mass spectroscopy. Fully deuterated methanol was used for confirmation of the results. Results. The photodesorption of methanol to the gas phase was not observed in the mass spectra at different irradiation temperatures. We estimate an upper limit of 3 × 10-5 molecules per incident photon. On the other hand, photon-induced desorption of the main photoproducts was clearly observed. Conclusions. The negligible photodesorption of methanol could be explained by the ability of UV-photons in the 114-180 nm (10.87-6.88 eV) range to dissociate this molecule efficiently. Therefore, the presence of gas-phase methanol in the absence of thermal desorption remains unexplained. On the other hand, we find CH4 to desorb from irradiated methanol ice, which was not found to desorb in the pure CH4 ice irradiation experiments.