Recently, I looked at articles that make use of Deligne's idea that "in characteristic 0 every deformation problem is governed by a differential graded
Lie algebra" as explained first in Goldman-Millson (1988).


So I wonder what precisely goes wrong in (mixed) characteristic p>2? 



Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?)
More precisely, the "Deligne principle" of deformation theory (but actually one could add a lot of other names) is that every deformation problem corresponds to a deformation functor, which in turn is defined by a certain dg Lie algebra controlling the deformations.
More precisely, from any dg Lie algebra $g$ and any commutative algebra $A$ you obtain a new dg Lie algebra $g\otimes\_{\mathbb{K}}A$, and this gives rise to a functor
$$
MC(g\otimes-):\textit{Com-Alg}\rightarrow \textit{Sets}
$$
which is your deformation functor.
Now, you would like to define equivalences classes of deformations over $A$ by a certain equivalence relation in $MC(g\otimes A)$. For this, you use the fact that the degree $0$ part of your dg Lie algebra $g\otimes A$ is a Lie algebra that you can exponentiate into an algebraic group $G(A)$ called the gauge group (under nilpotence assumption). The gauge group acts on the Maurer-Cartan elements, and the equivalences classes are given by the quotient under this action.


I think you should at least encounter some troubles if you try to exponentiate your Lie algebra in positive characteristic. As far as I know, you have to use exponential power series. Moreover, in positive characteristic $p$ you have to use restricted Lie algebras or Lie $p$-algebra (in order to take into account the operation coming from the Frobenius map),
and consider nilpotent.
I do not know any reference about deformation theory "à la Deligne" in positive characteristic using restricted Lie algebras.


