Suppose $m,n\geq 2$ are two integers. Is it true that for every sufficiently large nonabelian group $G$, one can find a set $A\subset G$, with $|A|=n$, so that $|A^m| >\binom{n+m-1}{m}$?

(Edit) Let's also add the condition $m\le n$ since the answer below provides a counter-example for large enough $m$. In general it would be interesting to know the range of $(m,n)$ for which the statement holds.(/Edit)

Here $A^k=\lbrace a_1a_2\cdots a_k| a_1,a_2,\dots,a_k\in A\rbrace$ is a product set. It is obvious that in every abelian group one has $|A^m| \le\binom{n+m-1}{m}$, for every $A$.

I don't have an application in mind, I was trying the case $m=2$ and I think I have a proof (still haven't checked all the steps, but it's not particularly enlightening since it splits into many cases). I'm wondering if this is true in general and if there is a slick proof, or if there is a counter-example.