  Let $G \leq \mathrm{GL}(V)$ 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.
