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We investigate the presence of COMs in strongly UV-irradiated interstellar molecular gas. We have carried out a complete millimetre line survey using the IRAM30m telescope towards the edge of the Orion Bar photodissociation region (PDR), close to the H2 dissociation front, a position irradiated by a very intense far-UV (FUV) radiation field. These observations have been complemented with 8.5 arcsec resolution maps of the H2CO 5(1,5)-4(1,4) and C18O 3-2 emission at 0.9 mm. Despite being a harsh environment, we detect more than 250 lines from COMs and related precursors: H2CO, CH3OH, HCO, H2CCO, CH3CHO, H2CS, HCOOH, CH3CN, CH2NH, HNCO, H13-2CO, and HC3N (in decreasing order of abundance). For each species, the large number of detected lines allowed us to accurately constrain their rotational temperatures (Trot) and column densities (N). Owing to subthermal excitation and intricate spectroscopy of some COMs (symmetric- and asymmetric-top molecules such as CH3CN and H2CO, respectively), a correct determination of N and Trot requires building rotational population diagrams of their rotational ladders separately. We also provide accurate upper limit abundances for chemically related molecules that might have been expected, but are not conclusively detected at the edge of the PDR (HDCO, CH3O, CH3NC, CH3CCH, CH3OCH3, HCOOCH3, CH3CH2OH, CH3CH2CN, and CH2CHCN). A non-LTE LVG excitation analysis for molecules with known collisional rate coefficients, suggests that some COMs arise from different PDR layers but we cannot resolve them spatially. In particular, H2CO and CH3CN survive in the extended gas directly exposed to the strong FUV flux (Tk = 150-250 K and Td > 60 K), whereas CH3OH only arises from denser and cooler gas clumps in the more shielded PDR interior (Tk = 40-50 K). We find a HCO/H2CO/CH3OH = 1/5/3 abundance ratio. These ratios are different from those inferred in hot cores and shocks.