I am trying to use Hensel's lemma to find all the elements which have a square root in $R=\mathbb{Z}_3[[j]]$. We consider $f(x)=x^2-\sum_{i=0}^{\infty}a_ij^i$ a monic polynomial in $R[x]$. Looking at this polynomial modulo the ideal $(3,j)$, we get $\tilde{f}(x)=x^2-a_0$ over $\mathbb{F}_3[x]$. This breaks into linear factors if and only if $a_0 \equiv 1 \bmod 3$.
Assuming $a_0 \equiv 1 \bmod 3$, by Hensel's lemma we can lift these factors to $R[x]$. So the only elements that have square roots are power series whose constant terms is congruent to $1$ modulo $3$.
I am not sure if I am applying Hensel's lemma correctly here. Any references or answers which justify or contradict this argument as really appreciated. Thank you.