The first part of the paper is dedicated to the construction of a nonparametric confidence interval for a conditional quantile of level depending on the sample size. When this level converges to 0 or 1 as the sample size increases, the conditional quantile is said to be extreme and is located in the tail of the conditional distribution. The proposed confidence interval is constructed by approximating the distribution of ordered statistics selected with a nearest neighbour approach by a Beta distribution. We show that its coverage probability converges to the preselected probability and its accuracy is illustrated on a simulation study. When the dimension of the covariate increases, the coverage probability of the confidence interval can be very different from the nominal one. This is a well known consequence of the data sparsity especially in the tail of the distribution. In the second part of the paper, a dimension reduction procedure is proposed in order to select more appropriate nearest neighbours and in turn to obtain a better coverage probability. This procedure is based on the Tail Conditional Independence assumption introduced in (Gardes, Extreme, pp. 57-95, 18(3), 2018).