Given the solution of a diophantine polynomial $f$ in $\mathbb{F}_p$, we can use Hensel's lemma to extend this to a solution in $\mathbb{Z}_p$, and the natural homomorphism $\mathbb{Z}_p \to \mathbb{Q}_p$ gives a solution in $p-$ adics. I heard it mentioned that we can go the other way, and I see how we can get a solution in $\mathbb{F}_p$ from a solution in $\mathbb{Z}_p$ using the direct limit construction, but I can't figure out how to get from $\mathbb{Q}_p$ to $\mathbb{Z}_p$ or to $\mathbb{F}_p$.
References or hints are appreciated!