I am somewhat confused about the following two concepts and the relations between them-
One concept is a Lie group $G$ over the $p$-adic field. This is defined in a similar fashion to a (real) Lie group - using atlases of charts, while making sure to work in the analytic (=defined via a power series) category.
The other concept is the group $H=\mathbb{H}(k)$ of $k$ points for a $k$-defined algebraic group $\mathbb{H}$, for $k=\mathbb{Q}_p$. As any algebraic group, $\mathbb{H}$ is a matrix group, and hence so is $H$ as well.
Every group of the second type can be considered as a group of the first kind under an appropriate atlas (as is the case over any local field).
On the other hand, for $k=\mathbb{R}$ for example it is known (if im not mistaken?) that every semisimple Lie group with trivial center, and every compact group is in fact the connected component of the $\mathbb{R}$-points of some algebraic group defined over $\mathbb{R}$.
My question is - do similar results hold over $\mathbb{Q}_p$? Obviously we cannot depend on taking the connected component... In other words, when is a $p$-adic Lie group in fact a algebraic? What about a semi-simple $p$-adic Lie group (i.e. having a semi-simple Lie algebra)?
Thank you in advance