A closed-form expression is obtained for a holomorphic sector of the two-loop low-energy effective action for the N=4 super Yang–Mills theory on its Coulomb branch where the gauge group SU(N) is spontaneously broken to SU(N−1)×U(1) and the dynamics is described by a single Abelian N=2 vector multiplet. In the framework of the background-field method, this holomorphic sector is singled out by computing the effective action for a background N=2 vector multiplet satisfying a relaxed super self-duality condition. At the two-loop level, the N=4 SYM effective action is shown to possess no F4 term (with F the U(1) field strength), in accordance with the Dine–Seiberg non-renormalization conjecture hep-th/9705057 and its generalized form given in hep-th/0310025. An unexpected outcome of our calculation is that no (manifestly supersymmetric invariant generating) F6 quantum correction occurs at two loops. This is in conflict with previous results.