I've been reading some papers on Igusa zeta functions, and they seem to be implicitly using a "quantitative version" of Hensel's Lemma, which also asserts that the number of lifts of a $\mathbb{Z}/p\mathbb{Z}$-point to a $\mathbb{Z}/p^k\mathbb{Z}$-point. I'm looking for something like the following:

> Let $X$ be a smooth irreducible separated scheme of finite type of relative dimension $n$ over the ring of $p$-adic integers $\mathbb{Z}_p$. Then for any $k>0$ do we have
$$\# X(\mathbb{Z}/p^k\mathbb{Z}) = p^{n(k-1)}\# X(\mathbb{Z}/p\mathbb{Z}) \quad ?$$

I'm looking for either a proof or a reference where I can find a proof.