I'm studying for a course in number theory and I have seen that:
$\Bbb Q_p$ is complete w.r.t. the $p$-adic norm
$\Bbb Q_p^{unram}$, the union of all unramified extensions of $\Bbb Q_p$, is not complete
But what if we take a single unramified extension? Is it complete? So, can we use Hensel's lemma to lift roots?
Every finite extension of $\Bbb Q_p$ is complete with respect to the extended norm. In general infinite algebraic extensions won't be.
In your example of $\Bbb Q_p^{\textrm{unram}}$ I would try looking at $\sum_{n=1}^\infty p^n \zeta_n$, where $\zeta_n$ is a root of unity in the degree $2^n$ unramified extension of $\Bbb Q_p$ but not in any smaller extension. Can you prove that this series doesn't converge in $\Bbb Q_p^{\textrm{unram}}$? If it did, it would converge to some element of a finite extension of $\Bbb Q_p$.