High levels of deuterium fractionation in gas-phase molecules are usually associated with cold regions, such as prestellar cores. Significant fractionation ratios are also observed in hot environments such as hot cores or hot corinos, where they are believed to be produced by the evaporation of the icy mantles surrounding dust grains, and thus are remnants of a previous cold (either gas-phase or grain surface) chemistry. The recent detection of DCN towards the Orion Bar, in a clump at a characteristic temperature of 70K, has shown that high deuterium fractionation can also be detected in PDRs. The Orion Bar clumps thus appear as a good environment for the observational study of deuterium fractionation in luke-warm gas, allowing to validate chemistry models in a different temperature range, where dominating fractionation processes are predicted to be different than in cold gas (< 20K). We aimed at studying observationally in detail the chemistry at work in the Orion Bar PDR, to understand if DCN is produced by ice mantle evaporation, or is the result of warm gas-phase chemistry, involving the CH2D+ precursor ion (which survives higher temperatures than the usual H2D+ precursor). Using the APEX and the IRAM 30m telescopes, we targetted selected deuterated species towards two clumps in the Orion Bar. We confirmed the detection of DCN and detected two new deuterated molecules (DCO+ and HDCO) towards one clump in the Orion Bar PDR. Significant deuterium fractionations are found for HCN and H2CO, but a low fractionation in HCO+. We also give upper limits for other molecules relevant for the deuterium chemistry. (...) We show evidence that warm deuterium chemistry driven by CH2D+ is at work in the clumps.