We show that the restricted Lie algebra structure on Hochschild cohomology is invariant under stable equivalences of Morita type between self-injective algebras. Thereby, we obtain a number of positive characteristic stable invariants, such as the p-toral rank of HH1 (A, A). We also prove a more general result concerning Iwanaga–Gorenstein algebras, using a generalization of stable equivalences of Morita type. Several applications are given to commutative algebra and modular representation theory. These results are proven by first establishing the stable invariance of the B∞-structure of the Hochschild cochain complex. In the appendix, we explain how the p-power operation on Hochschild cohomology can be seen as an artifact of this B∞-structure. In particular, we establish well-definedness of the p-power operation, following some—originally topological—methods due to May, Cohen and Turchin, using the language of operads.