Summary: The purpose of this note is to add to the evidence that the algebra structure of a \(p\)-block of a finite group is closely related to the Lie algebra structure of its first Hochschild cohomology group. We show that if \(B\) is a block of a finite group algebra \(kG\) over an algebraically closed field \(k\) of prime characteristic \(p\) such that \(HH^1(B)\) is a simple Lie algebra and such that \(B\) has a unique isomorphism class of simple modules, then \(B\) is nilpotent with an elementary abelian defect group \(P\) of order at least 3, and \(HH^1(B)\) is in that case isomorphic to the Witt algebra \(HH^1(kP)\). In particular, no other simple modular Lie algebras arise as \(HH^1(B)\) of a block \(B\) with a single isomorphism class of simple modules.