Artin/Popescu approximation for (some) big rings

Fix a prime number $p$. Let $A = \overline{\mathbf{Z}_p}$ be the integral closure of the $p$-adic integers $\mathbf{Z}_p$ in some fixed algebraic closure of its fraction field, and let $B$ be the $p$-adic completion of $A$. Is the map $A \to B$ an inductive limit of smooth morphisms? If $A$ was excellent, this would follow from Artin/Popescu approximation theorems, but $A$ is not even noetherian. Of course, one can ask similar questions much more generally, but this is the case I am interested in.

flag	asked Sep 11 at 16:23 anon 402●2●8

	It may be helpful if you indicate why you pose the question (e.g., idle curiosity or something more substantial). For example, some non-archimedean geometry (most naturally, Berkovich spaces) ensures that for a finite system of polynomial equations over $A$ (or even something more general), any solution in $B$ can be approximated arbitrarily well by a solution in $A$. So if that is your goal then the Artin-Popescu (and "smoothening") viewpoint in such generality is unnecessary. – grp Sep 12 at 4:25
	I would like to know what the cotangent complex of $B$ relative to $\mathbf{Z}_p$ is. If the question I ask has an affirmative answer, then the relative cotangent complex of $B$ over $A$ is concentrated in degree $0$ (and is moreover a $\mathbf{Q}_p$-module). – anon Sep 13 at 12:42