Elkik in Solutions d'equations a coefficients dans un anneu Henselian, Theorem 7 proves that:

Let $A$ be a Noetherian ring that is Henselian with respect to a principal ideal $(a)$.

That is, if $f(x)$ is a monic polynomial with a root $\alpha \in A/(a)$ such that $df/dx (\alpha)$ is a unit in $A/(a)$, then $\alpha$ lifts to a root of $f$ in $A$.

Let $\hat{A}$ be its $a$-adic completion, and let $\hat{B}$ be a formally finitely generated algebra over $\hat{a}$ that is formally smooth over $\hat{A}[a^{-1}]$. Then there exists a finitely generated algebra $B$ over $A$ that is smooth over $A[a^{-1}]$ and such that its $a$-adic completion is isomorphic to $\hat{B}$.

Is this theorem true in the non-Noetherian case? If not, what is a counterexample?

I would ideally prefer counterexamples of relative dimension $0$ such as finite etale covers.