This is a question in Niven's An Introduction to the Theory of Numbers.
I believe a result from the previous exercise
If $p\geq 5$ and $m$ is a positive integer then $\binom{mp-1}{p-1} \equiv 1 \pmod {p^{3}}$
would help, and I had tried to apply Hensel's Lemma to it. But I failed to construct a function that can link the two questions.
Another approach that I had come up with is by induction on the positive integer $m$.
First for the base case where $m=1$, the result is trivial.
And for the induction step I assume that the proposition holds for $k=m-1$. Then for $k=m$, note that $\frac{(mp)!}{(p)![(m-1)p]!}=\binom{mp}{p}$ , and by the induction hypothesis we have $[(m-1)p]! \equiv (m-1)!p!^{m-1} \pmod{p^{m+2}}$, but I cannot deduce its remainder modulo $p^{m+3}$ from the hypothesis. So the method may comes to a dead end.