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Astrophysical observations show complex organic molecules (COMs) in the gas phase of protoplanetary disks. X-rays emitted from the central young stellar object (YSO) that irradiate interstellar ices in the disk, followed by the ejection of molecules in the gas phase, are a possible route to explain the abundances observed in the cold regions. This process, known as X-ray photodesorption, needs to be quantified for methanol-containing ices. We aim at experimentally measuring X-ray photodesorption yields of methanol and its photo-products from binary mixed ices: $^{13}$CO:CH$_3$OH ice and H$_2$O:CH$_3$OH ice. We irradiated these ices at 15 K with X-rays in the 525 - 570 eV range. The release of species in the gas phase was monitored by quadrupole mass spectrometry, and photodesorption yields were derived. For $^{13}$CO:CH$_3$OH ice, CH$_3$OH X-ray photodesorption yield is estimated to be 10$^{-2}$ molecule/photon at 564 eV. X-ray photodesorption of larger COMs, which can be attributed to either ethanol, dimethyl ether, and/or formic acid, is detected with a yield of 10$^{-3}$ molecule/photon. When methanol is mixed with water, X-ray photodesorption of methanol and of the previous COMs is not detected. X-ray induced chemistry, dominated by low-energy secondary electrons, is found to be the main mechanism that explains these results. We also provide desorption yields that are applicable to protoplanetary disk environments for astrochemical models. The X-ray emission from YSOs should participate in the enrichment of the protoplanetary disk gas phase with COMs such as methanol in the cold and X-ray dominated regions because of X-ray photodesorption from methanol-containing ices., Comment: arXiv admin note: text overlap with arXiv:2101.06179