The étale homotopy type is a construction due to Artin and Mazur that generalizes the étale fundamental group. If $X$ is a scheme over a separably closed field $k$, then the étale homotopy type of $X$ is a profinite homotopy type that knows everything about the étale cohomology of $X$ with finite coefficients. If the base field is $\mathbb{C}$, and $X$ is not too bad, the etale homotopy type of $X$ is just the profinite completion of the topological space that underlies the set of $\mathbb{C}$-points of $X$. I am wondering what is known if $k$ is of positive characteristic ?

Here is a precise question : Are there examples of smooth and proper schemes $X$ over the algebraic closure of $\mathbb{F}_{p}$ that are such that the étale homotopy type of $X$ is not that of a complex manifold if we complete away from $p$ ? I am also willing to drop the assumption that $X$ is proper if this makes the question more interesting.