Estimation of the extreme-value index of a heavy-tailed distribution is addressed when some random covariate information is available and the data are randomly right-censored. An inverse-probability-of-censoring-weighted kernel version of Hill's estimator of the extreme-value index is proposed and its asymptotic normality is established. Based on this, a Weissman-type estimator of conditional extreme quantiles is also constructed. A simulation study is conducted to assess the finite-sample behaviour of the proposed estimators.