Complex organic molecules ( COMs ) have been detected in a variety of environments , including cold prestellar cores . Given the low temperature of these objects , these last detections challenge existing models . We report here new observations towards the prestellar core L1544 . They are based on an unbiased spectral survey of the 3mm band at the IRAM-30m telescope , as part of the Large Program ASAI . The observations allow us to provide the 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 \times 10 ^ { -11 } to 6 \times 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 \sim 8000 AU where T \sim 10 K and n _ { H _ { 2 } } \sim 2 \times 10 ^ { 4 } cm ^ { -3 } . Previous works have shown that water vapour 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 tiny 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 .