We consider a general multivariate affine stochastic recursion and the associated Markov chain on $\mathbb R^{d}$. We assume a natural geometric condition which implies existence of an unbounded stationary solution and we show that the large values of the associated stationary process follow extreme value properties of classical type, with a non trivial extremal index. We develop some explicit consequences such as convergence to Fr{é}chet's law or to an exponential law, as well as convergence to a stable law. The proof is based on a spectral gap property for the action of associated positive operators on spaces of regular functions with slow growth, and on the clustering properties of large values in the recursion.