Let G≤GL(V) be a group with a unique subnormal quasisimple subgroup E(G) that acts absolutely irreducibly on V. A base for G acting on V is a set of vectors with trivial pointwise stabiliser in G. In this paper we determine the minimal base size of G when E(G)/Z(E(G)) is a finite simple group of Lie type in cross-characteristic. We show that G has a regular orbit on V, with specific exceptions, for which we find the base size.