Wavelength dependent photodesorption rates have been determined using synchrotron radiation, for condensed pure and mixed methanol ice in the 7 -- 14 eV range. The VUV photodesorption of intact methanol molecules from pure methanol ices is found to be of the order of 10$^{-5}$ molecules/photon, that is two orders of magnitude below what is generally used in astrochemical models. This rate gets even lower ($<$ 10$^{-6}$ molecules/photon) when the methanol is mixed with CO molecules in the ices. This is consistent with a picture in which photodissociation and recombination processes are at the origin of intact methanol desorption from pure CH$_3$OH ices. Such low rates are explained by the fact that the overall photodesorption process is dominated by the desorption of the photofragments CO, CH$_3$, OH, H$_2$CO and CH$_3$O/CH$_2$OH, whose photodesorption rates are given in this study. Our results suggest that the role of the photodesorption as a mechanism to explain the observed gas phase abundances of methanol in cold media is probably overestimated. Nevertheless, the photodesorption of radicals from methanol-rich ices may stand at the origin of the gas phase presence of radicals such as CH$_3$O, therefore opening new gas phase chemical routes for the formation of complex molecules.
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