Consider the following diagram of regular local rings

$\begin{matrix} \hat{A} & \xrightarrow{\quad\hat\varphi\quad} & \hat{B} \\ \ \uparrow\scriptstyle\alpha & \circlearrowleft & \ \uparrow\scriptstyle\beta \\ A & \xrightarrow{\quad\varphi\quad} & B \end{matrix}$

where $\widehat{\\,\dot\\,}$ denotes the completion functor. Let $m\subset A$ and $n\subset B$ be the respective maximal ideals. Assume that $\varphi$ is injective and makes $B$ integral over $A$, in particular all morphisms are inclusions.

Given $\hat y\in\hat n\setminus \hat n^2$ such that $\hat y^k=x \in m\setminus m^2$, can I find $y\in B$ such that $y^k = x$?