This question is related to Lifting points via étale morphism of adic spaces.

Fix a complete non-archimedean field $ k$ . Let $ (A,A^+)$ be a complete strongly noetherian Huber pair over $ (k,k^\circ)$ . Let $ f\colon A\to B$ be a finite étale morphism of rings; endow $ B$ with the natural $ A$ -module topology and let $ B^+$ be the integral closure of $ A^+$ in $ B$ . Then $ f$ defines a morphism of complete Huber pairs $ f\colon (A,A^+)\to (B,B^+)$ over $ (k,k^\circ)$ .

Let now $ x\in \operatorname{Spa}(A,A^+)$ and suppose that $ \operatorname{Spa}(k(x),k(x)^+)\to X:=\operatorname{Spa}(A,A^+)$ lifts to $ Y:=\operatorname{Spa}(B,B^+)$ . Denote the corresponding point in $ Y$ by $ y$ . Then the inclusion $ k(x)\subset \widehat{k(x)}$ factors through $ k(y)$ : $ $ k(x)\subset k(y)\subset \widehat{k(x)}. $ $ Thus, if we assume that $ k(x)$ is already complete, we obtain $ k(x)=k(y)$ . Now the morphism $ f_y\colon \mathcal{O}_{X,x}\to \mathcal{O}_{Y,y}$ induced by $ f$ is étale at the closed point of $ \mathcal{O}_{Y,y}$ . Since $ \mathcal{O}_{X,x}$ is henselian, the equality $ k(x)=k(y)$ implies that $ f_y$ has a section $ s\colon \mathcal{O}_{Y,y}\to \mathcal{O}_{X,x}$ , so that we obtain a map $ s\colon B\to \mathcal{O}_{X,x}$ . The noetherian assumption implies that this lifts to a morphism $ s\colon B\to \mathcal{O}_X(U)$ , for $ U$ a rational subset of $ X$ containing $ x$ .

In other words, the morphism $ f\colon Y\to X$ admits a section $ s\colon U\to Y$ locally around $ x$ .

$ \textbf{Question}$ : How can I argue if the residue field $ k(x)$ is not necessarily complete?

Help is highly appreciated.