Let $R$ be a smooth, integral, finite-type $\mathbb{Z}_{(p)}$-algebra of relative dimension $n$ and $\overline{f} \colon R \to \mathbb{F}_p$. Then Hensel's lemma tells us that this lifts to a map $R \to \mathbb{Z}_p$. My understanding is that the space of lifts looks like an affine space, but I would like to understand this more explicitly.

I'm in particular hoping that it's always possible to choose an injective lift. In geometric terms, this is asking for a $\mathbb{Z}_p$-point such that the associated $\mathbb{Q}_p$-point maps to the generic point of $X:=\mathrm{Spec}(R)$.

I'm hoping this has something to do with the tangent space - like we want a tangent vector whose coordinates in $\mathbb{Z}_p$ are algebraically independent over $\mathbb{Q}$. But I don't understand this deformation space.

I'm listed this as "reference-request" because it might be fairly standard from deformation theory, but I couldn't find a reference and don't know where to look.