If we want to find the roots of a polynomial $f(x)$ modulo a prime $p$ to the power of $n$, we can use Hensel's lemma. Let's say we want to find all roots of $x^3+x^2+4x+1$ mod $49$. Then we can use Hensel's lemma for p-adic integers and compute the roots only mod $7$.
Is there also a version of Hensel's lemma, not only for the completion of $\mathbb{Q}$, but for the completion of $\mathbb{F}_q(t)$? Or exists any other root lifting technique in this case? I didn't found anything so far...