<p>Extreme quantiles are standard risk measures in financial or environmental contexts since they provide information on low-probability but high-impact events. The calculation of accurate confidence intervals on extreme quantiles is then of paramount importance to risk managers. The celebrated Weissman estimator is a simple way to compute such extreme quantiles from heavy-tailed distributions. Asymptotic confidence intervals can also be built based on Weissman estimator asymptotic normality, but they may suffer from poor coverage properties in practice due to oversimplified estimations of the asymptotic bias and variance. We propose several higher order approximations of the Weissman estimator asymptotic distribution together with a data-driven procedure to automatically select the most appropriate one thanks to a supervised classification method. The discrepancy between theoretical and empirical coverages is studied using a new criterion robust with respect to the choice of the number of tail observations involved in the Weissman estimator. The usefulness of the associated adaptive confidence interval is illustrated on an intensive simulation study as well as on two climatic and financial data sets. The source code is available at https://github.com/antoineinria/adaptative_ci_code.</p>