  The purpose of this paper is to develop a deformation theory controlled by
pre-Lie algebras with divided powers over a ring of positive characteristic. We
show that every differential graded pre-Lie algebra with divided powers comes
with operations, called weighted braces, which we use to generalize the
classical deformation theory controlled by Lie algebras over a field of
characteristic $0$. Explicitly, we define the Maurer-Cartan set, as well as the
gauge group, and prove that there is an action of the gauge group on the
Maurer-Cartan set. This new deformation theory moreover admits a
Goldman-Millson theorem which remains valid on the integers. As an application,
we give the computation of the $\pi_0$ of a mapping space
$\text{Map}(B^c(\mathcal{C}),\mathcal{P})$ with $\mathcal{C}$ and $\mathcal{P}$
suitable cooperad and operad respectively.
