Since $g$ is adic it is representable by algebraic spaces (Formal Spaces, Section \ref{formal-spaces-section-adic}). Thus by saying $g$ is smooth we mean that $g$ should be smooth in the sense of Bootstrap, Definition \ref{bootstrap-definition-property-transformation}. \medskip\noindent Write $W = \colim W_n$ as in Formal Spaces, Lemma \ref{formal-spaces-lemma-structure-locally-noetherian}. Set $V_n = V_{/Z} \times_{\hat x, W} W_n$. Then $V_n$ is a closed subscheme with underlying set $Z$. Smoothness of $V \to W$ is equivalent to the smoothness of all the morphisms $V_n \to W_n$ (this holds because any morphism $T \to W$ with $T$ a quasi-compact scheme factors through $W_n$ for some $n$). We know that the morphism $V_n \to W_n$ is smooth at $v$ by Lemma \ref{lemma-base-change-versal}\footnote{The lemma applies since the diagonal of $W$ is representable by algebraic spaces and locally of finite type, see Formal Spaces, Lemma \ref{formal-spaces-lemma-diagonal-morphism-formal-algebraic-spaces} and we have seen that $W$ has (RS) in the proof of Lemma \ref{lemma-rs}.}. Of course this means that given any $n$ we can shrink $V$ such that $V_n \to W_n$ is smooth. The problem is to find an open which works for all $n$ at the same time. \medskip\noindent The question is local on $V$, hence we may assume $S = \Spec(R)$ and $V = \Spec(A)$ are affine. \medskip\noindent In this paragraph we reduce to the case where $W$ is an affine formal algebraic space. Choose an affine formal scheme $W'$ and an \'etale morphism $W' \to W$ such that the image of $v$ in $W_{red}$ is in the image of $W'_{red} \to W_{red}$. Then $V_{/Z} \times_{g, W} W' \to V_{/Z}$ is an adic \'etale morphism of formal algebraic spaces over $S$ and $V_{/Z} \times_{g, W} W'$ is an affine fromal algebraic space. By Algebraization of Formal Spaces, Lemma \ref{restricted-lemma-algebraize-rig-etale-affine} there exists an \'etale morphism $\varphi : V' \to V$ of affine schemes such that the completion of $V'$ along $Z' = \varphi^{-1}(Z)$ is isomorphic to $V_{/Z} \times_{g, W} W'$ over $V_{/Z}$. Observe that $v$ is the image of some $v' \in V'$. Since smoothness is preserved under base change we see that $V'_n \to W'_n$ is smooth for all $n$. In the next paragraph we show that after replacing $V'$ by an open neighbourhood of $v'$ the morphisms $V'_n \to W'_n$ are smooth for all $n$. Then, after we replace $V$ by the open image of $V' \to V$, we obtain that $V_n \to W_n$ is smooth by \'etale descent of smoothness. Some details omitted. \medskip\noindent Assume $S = \Spec(R)$, $V = \Spec(A)$, $Z = V(I)$, and $W = \text{Spf}(B)$. Let $v$ correspond to the maximal ideal $I \subset \mathfrak m \subset A$. We are given an adic continuous $R$-algebra homomorphism $$ B \longrightarrow A^\wedge $$ Let $\mathfrak b \subset B$ be the ideal of topologically nilpotent elements (this is the maximal ideal of definition of the Noetherian adic topological ring $B$). Observe that $\mathfrak b A^\wedge$ and $IA^\wedge$ are both ideals of definition of the Noetherian adic ring $A^\wedge$. Also, $\mathfrak m A^\wedge$ is a maximal ideal of $A^\wedge$ containing both $\mathfrak b A^\wedge$ and $IA^\wedge$. We are given that $$ B_n = B/\mathfrak b^n \to A^\wedge/\mathfrak b^n A^\wedge = A_n $$ is smooth at $\mathfrak m$ for all $n$. By the discussion above we may and do assume that $B_1 \to A_1$ is a smooth ring map. Denote $\mathfrak m_1 \subset A_1$ the maximal ideal corresponing to $\mathfrak m$. Since smoothness implies flatness, we see that: for all $n \geq 1$ the map $$ \mathfrak b^n/\mathfrak b^{n + 1} \otimes_{B_1} (A_1)_{\mathfrak m_1} \longrightarrow \left(\mathfrak b^nA^\wedge/\mathfrak b^{n + 1}A^\wedge\right)_{\mathfrak m_1} $$ is an isomorphism (see Algebra, Lemma \ref{algebra-lemma-what-does-it-mean-again}). Consider the Rees algebra $$ B' = \bigoplus\nolimits_{n \geq 0} \mathfrak b^n/\mathfrak b^{n + 1} $$ which is a finite type graded algebra over the Noetherian ring $B_1$ and the Rees algebra $$ A' = \bigoplus\nolimits_{n \geq 0} \mathfrak b^nA^\wedge/\mathfrak b^{n + 1}A^\wedge $$ which is a a finite type graded algebra over the Noetherian ring $A_1$. Consider the homomorphism of graded $A_1$-algebras $$ \Psi : B' \otimes_{B_1} A_1 \longrightarrow A' $$ By the above this map is an isomorphism after localizing at the maximal ideal $\mathfrak m_1$ of $A_1$. Hence $\Ker(\Psi)$, resp.\ $\Coker(\Psi)$ is a finite module over $B' \otimes_{B_1} A_1$, resp.\ $A'$ whose localization at $\mathfrak m_1$ is zero. It follows that after replacing $A_1$ (and correspondingly $A$) by a principal localization we may assume $\Psi$ is an isomorphism. (This is the key step of the proof.) Then working backwards we see that $B_n \to A_n$ is flat, see Algebra, Lemma \ref{algebra-lemma-what-does-it-mean-again}. Hence $A_n \to B_n$ is smooth (as a flat ring map with smooth fibres, see Algebra, Lemma \ref{algebra-lemma-flat-fibre-smooth}) and the proof is complete.