Pro-finite completion of p-adic Lie groups

Question

Consider a $p$-adic Lie group $G$. My question is if the pro-finite completion $\hat{G}$ is a $p$-adic Lie group. First we note that since $$\hat{G}=\text{lim}_{N\subset G} G/N$$ where the limit ranges over all normal subgroups of finite index, and each $G/N$ is $p$-adic Lie group. So therefore the question reduces to the question if such a limit is a $p$-adic Lie group.

Answer 1

If we take the multiplicative group $G= \mathbb Q_p^\times$, then $\hat G \simeq \widehat{\mathbb Z} \times \mathbb Z_p^\times$ where the first factor is the (additive group of the) profinite completion of $\mathbb Z$ (by local CFT, this group is actually isomorphic to the Galois group of the maximal abelian extension $\mathbb Q_p^{ab} \mid \mathbb Q_p$ and hence of central interest). That first factor is well known to be isomorphic to the direct product of all additive groups of the $\ell$-adic integers for all (!) primes $\ell$, i.e. we get

$$\hat G \simeq \mathbb Z_p^\times \times\prod_{\ell \text{ prime}} \mathbb Z_\ell $$

Now I cannot shake a rigorous proof out of my sleeve right now, but I would be very surprised if this thing (well, the part $\prod_{\ell \text{ prime} \neq p} \mathbb Z_\ell$) is a $p$-adic Lie group.

On the other hand, I have a strong feeling that for compact $G$, we might be more lucky via the sources given in the comments.