We present a nonparametric family of estimators for the tail index of a Weibull tail-distribution when functional covariate is available. Our estimators are based on a kernel estimator of extreme conditional quantiles, extending a previous work Daouia et al. (2013) to the infinite dimensional case. Asymptotic normality of the estimators is proved under mild regularity conditions. Their finite sample performances are illustrated both on simulated and real data. We refer to Gardes and Girard (2016) for further details. Daouia, A., Gardes, L., Girard, S. (2013). On kernel smoothing for extremal quantile regression. Bernoulli, 19, 2557–2589. Gardes, L., Girard, S. (2016). On the estimation of the functional Weibull tail-coefficient, Journal of Multivariate Analysis, to appear, http://dx.doi.org/10.1016/j.jmva.2015.05.007