Given a nilpotent Lie algebra over a characteristic zero field, one can construct a group in a universal way via the Baker-Campbell-Hausdorff formula. This integration procedure admits generalizations to dg Lie or L∞-algebras, giving in general ∞-groupoid of deformations that it encodes, as by the Lurie-Pridham correspondence, infinitesimal deformation problems are equivalent to dg Lie algebras. The recent work of Brantner-Mathew establishes a correspondence between infinitesimal deformation problems and partition Lie algebras over a positive characteristic field. In this talk, I will explain how to construct an analogue of the integration functor for certain point-set models of (spectral) partition Lie algebras, and how this integration functor can recover the associated deformation problem under some assumptions. Furthermore, I will discuss some applications of these constructions to unstable p-adic homotopy theory.[-]