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Understanding how tree species will respond to a future climate requires reliable and quantitative estimates of intra‐specific variation under current climate conditions. We studied three 10‐year‐old common garden experiments established across a rainfall and drought gradient planted with nearly 10,000 pedunculate oak (Quercus robur L.) trees from ten provenances with known family structure. We aimed at disentangling adaptive and plastic responses for growth (height and diameter at breast height) as well as for leaf and wood functional traits related to adaptation to dry environments. We used restricted maximum likelihood approaches to assess additive genetic variation expressed as narrow‐sense heritability (h2), quantitative trait differentiation among provenances (QST), and genotype‐by‐environment interactions (GxE). We found strong and significant patterns of local adaptation in growth in all three common gardens, suggesting that transfer of seed material should not exceed a climatic distance of approximately 1°C under current climatic conditions, while transfer along precipitation gradients seems to be less stringent. Moreover, heritability reached 0.64 for tree height and 0.67 for dbh at the dry margin of the testing spectrum, suggesting significant additive genetic variation of potential use for future selection and tree breeding. GxE interactions in growth were significant and explained less phenotypic variation than origin of seed source (4% versus 10%). Functional trait variation among provenances was partly related to drought regimes at provenances origins but had moderate explanatory power for growth. We conclude that directional selection, either naturally or through breeding, is the most likely and feasible outcome for pedunculate oak to adapt to warmer and drier climate conditions in the future. [ABSTRACT FROM AUTHOR]