@stupidboy, Yes you are right, I was not careful. Also, Barnea and Larsen's result is not needed above because $F_r$, being of infinte rank, cannot embedded in $GL_n(\mathbb{Z}_p)$. Nevertheless, the situation is not much different if we want to argue for the general case. All can be reduced to one universal group!

I think they should! Barnea and Larsen proved that non-abelain free pro-$p$ groups are not linear over non-archimedian local fields, but $G$ being $p$-adic analytic is linear over $\mathbb{Z}_p$ and hence over $\mathbb{Q}_p$. It follows that a presentation of $G$ as a quotient of $F_r$, for some $r$, has a non-trivial kernel.

@Dave Benson, well, thanks for the remark and excuse me for that ! I was just too spontaneous with my thoughts; should I reedit the question according to your comment?

Thank you so much and sorry for the delay. Quite elementary and ingenious argument. You mean that $k$ is the minimal degree of a possible identity, as $k$ can have arbitray large values (for instance consider any multiple of $qp$).

$x^a=a^{-1}xa$, is this sufficient? In your comment, you mean $x^{a_1}\dots x^{a_n}=1$,...

It can be embedded in $GL_n(\mathbb{Z}_p)$, for some $n$ (such a group is $p$-adic analytic).