Let \(G\) be a finite almost simple group with socle \(G_0\). A transitive action of \(G\) on a set \(\Omega\) is said to be standard if either \(G_0\) is an alternating group \(A_n\) and \(\Omega\) is an orbit of subsets or partitions of \(\{1,\dots,n\}\) (up to equivalence) or \(G\) is a classical group in a subspace action (up to equivalence). Let \(b(G)\) be the minimal size of a base for \(G\).
The main result of the paper is Theorem 1.1: Let \(G\) be a finite almost simple classical group in a faithful primitive non-standard action. Then either \(b(G)\leq 4\) or \(G=U_6(2)\), the stabilizer of a point \(\omega\in\Omega\) is \(G_\omega\cong U_4(3)\cdot 2^2\) and \(b(G)=5\).