Complex organic molecules (COMs) have been detected in a variety of environments including cold prestellar cores. Given the low temperatures of these objects, these detections challenge existing models. We report here new observations toward the prestellar core L1544. They are based on an unbiased spectral survey of the 3 mm band at the IRAM 30 m telescope as part of the Large Program ASAI. The observations allow us to provide a full census of the oxygen-bearing COMs in this source. We detected tricarbon monoxide, methanol, acetaldehyde, formic acid, ketene, and propyne with abundances varying from 5 × 10^-11 to 6 × 10^-9. The non-LTE analysis of the methanol lines shows that they are likely emitted at the border of the core at a radius of ~8000 AU, where T ~ 10 K and n _{H_2} ~2 × 10⁴ cm^-3. Previous works have shown that water vapor is enhanced in the same region because of the photodesorption of water ices. We propose that a non-thermal desorption mechanism is also responsible for the observed emission of methanol and COMs from the same layer. The desorbed oxygen and a small amount of desorbed methanol and ethene are enough to reproduce the abundances of tricarbon monoxide, methanol, acetaldehyde, and ketene measured in L1544. These new findings open the possibility that COMs in prestellar cores originate in a similar outer layer rather than in the dense inner cores, as previously assumed, and that their formation is driven by the non-thermally desorbed species.