A Dubnov-Ivanov-Nagata-Higman Theorem for NIP nilrings

We prove that NIP nilrings of bounded nilexponent are nilpotent. We also deduce that NIP $\omega$-categorical $n$-Engel Lie algebras of characteristic $p>n$ are nilpotent, yielding a case of the Lie algebra analogous of Wilson conjecture for $\omega$-categorical groups.