A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrary positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the $d$-dimensional torus and are connected to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering we can independently tune the power law exponent $τ$ of the degree distribution and the rate $δd$ at which the connection probability decreases with the distance of two vertices. We show that the network is robust if $τ<2+1/δ$, but fails to be robust if $τ>3$. In the case of one-dimensional space we also show that the network is not robust if $τ<2+1/(δ-1)$. This implies that robustness of a scale-free network depends not only on its power-law exponent but also on its clustering features. Other than the classical models of scale-free networks our model is not locally tree-like, and hence we need to develop novel methods for its study, including, for example, a surprising application of the BK-inequality.