When does there exist a section of $GL_n(\mathbb Z_p) \rightarrow GL_n(\mathbb F_p)$?

There is a reduction map $f: GL_n(\mathbb Z_p) \rightarrow GL_n(\mathbb F_p)$ for any prime $p$, when does there exist a group homomorphism $g: GL_n(\mathbb F_p) \rightarrow GL_n(\mathbb Z_p)$ such that $f\circ g=id$?

If $n=1$ this is always possible, and if $p-1>n$ this is always impossible as $\mathbb F_p \not \subseteq GL_n(\mathbb Z_p)$ by considering the minimal polynomial.

Unless there are some small counterexamples (like maybe n=p=2?) that I don't want to account for, the answer is never.

$GL_n(\mathbb{Z}_p)$ can be embedded inside $GL_n(\mathbb{C})$ (as $\mathbb{Z}_p$ can be embedded as a ring in $\mathbb{C}$), but (away from a few small pairs (n,p)) the smallest nontrivial complex representation of $SL_n(\mathbb{F}_p)$ has dimension $\frac{p^n - p}{p-1} > n$. See "Minimal characters of the finite classical groups" by Tiep and Zalesski for a chart of low dimensional characters.