Voisin examples in $p$-adic geometry Let $K$ be an algebraic closure of p-adic rationals. Does there exist a proper smooth rigid-analytic variety over $K$ whose etale homotopy type is not isomorphic to etale homotopy type of a proper smooth scheme over $K$? 
 A: Welcome new contributor.  There is an analogue of the Hopf surface in rigid analytic geometry.  Here is one link to an article about these.
MR1149806 (93g:32048) 
Voskuil, Harm 
Non-Archimedean Hopf surfaces. 
Sém. Théor. Nombres Bordeaux (2) 3 (1991), no. 2, 405–466. 
http://www.numdam.org/article/JTNB_1991__3_2_405_0.pdf
In particular, since the (first) homotopy group of these rigid analytic spaces is Abelian of odd rank, they cannot be "homotopic" to a proper smooth scheme over $K$.
This was not asked, but I will relate the following anyway.  Over the rigid analytic space of the projective line, there exists a proper smooth morphism to the projective line whose fibers are these rigid Hopf surfaces such that the total space is "rationally connected" in the sense that any pair of points are contained in a rigid analytic subspace that is isomorphic to the projective line.  Yet the total space admits a free action of a finite cyclic group.  Via "twisting", this gives a "rationally connected" fibration over a rigid analytic curve that has no section (not even topologically).  
Thus, just as the usual Hopf surfaces show that there can be no complex-analytic analogue of the "Rationally Connected Fibration Theorem", also these rigid analytic Hopf surfaces prove that also there is no rigid-analytic analogue.
