In rational homotopy theory, one can study the rational homotopy and cohomology category via an algebraic structure, respectively the rational Lie Algebra model and the Sullivan cdga model.

If I am not wrong, over a prime there is a similar theorem for $\mathbb{F}_p$ cohomology, that is Mandell's Theorem: under adequate assumptions, $\mathbb{F}_p$ cohomology together with its $E_{\infty}$ structure does a job analogous to Sullivan cdga in rational homotopy theory.

I am definitely not confident in this field, but I don't understand if $p$-homotopy groups $\mathbb{F}_p \otimes \pi_n$ (or some $p$-adic friend) are determined by Mandell's model or not, and what is the analog of the Lie model.

What I want in the end is a generalization of the notion of coformality to the $p$-adic world, which I recall: $X$ is said to be conformal if there exists a (zig-zag of) quasi-isomorphism connecting the rational homotopy groups of $\Omega X$ with the free Lie Algebra on $C^* X$.