The dependence structure of max-stable random vectors can be characterized by their Pickands dependence function. In many applications, the extremal dependence measure varies with covariates. We develop a flexible, semi-parametric method for the estimation of non-stationary multivariate Pickands dependence functions. The proposed construction is based on an accurate max-projection allowing to pass from the multivariate to the univariate setting and to rely on the generalized additive modeling framework. In the bivariate case, the resulting estimator of the Pickands function is regularized using constrained median smoothing B-splines, and bootstrap variability bands are constructed. In higher dimensions, we tailor our approach to the estimation of the extremal coefficient. An extended simulation study suggests that our estimator performs well and is competitive with the standard estimators in the absence of covariates. We apply the new methodology to a temperature dataset in the US where the extremal dependence is linked to time and altitude.