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.

This is not an answer to the question in the main body, but the one raised by the OP in the comments. The goal is to show that $L_{B/A}$ is concentrated in degree $0$ and $\mathbf{Q}_p$module for $A = \overline{\mathbf{Z}_p}$ and $B$ the $p$adic completion of $A$. The map $A \to B$ is flat and an isomorphism after reduction modulo $p$. The flat base change formula for the cotangent complex shows $$L_{B/A} \otimes_B^L B/p \simeq 0,$$ i.e., that $p$ acts invertibly on $L_{B/A}$. By flat base change again, we get $$L_{B/A} = L_{B/A}[p^{1}] \simeq L_{B[p^1]/A[p^{1}]}.$$ As $A[p^{1}] \to B[p^{1}]$ is an extension of fields of characteristic $0$, it is enough to show: Claim: If $E/K$ is an extension of fields of characteristic $0$, then $L_{E/K}$ is an $E$vector space placed in degree $0$. Proof: By expressing $E$ as a filtered colimit of finitely generated field extensions, as the cotangent complex commtues with such colimits, we may assume $E/K$ is finitely generated, i.e., there exists a finitely generated $K$algebra $A$ which is a domain such that $E$ is the fraction field of $A$. By generic smoothness (as we are in characteristic $0$), we may assume $A$ is smooth over $K$. Then $L_{A/K} = \Omega^1_{A/K}$ is a finite locally free $A$module in degree $0$. Localising at the generic point and using $L_{A/K} \otimes_A E \simeq L_{E/K}$ then proves the claim. 


Let me answer (?) your question with a question. What is this question really asking? It seems that it asks something like this. Let's call an Avariety a finitely presented, irreducible, scheme X over A. Now, suppose that we have a Bpoint x ∈ X(B). Assume that the image of x is dense in X. Question: Can we find a morphism Y > X of Avarieties and a Bpoint y of Y which maps to X. OK, and we can certainly find an alteration Y > X such that Y is the base change of a strictly semistable scheme over a suitable dvr, say R, contained in A. Since Y > X is an alteration and since B is algebraically closed (right?) we can lift x to a y in Y(B). Right? This already proves something about your cotangent complex because Y is lci over A. Right? 

