Every group of totally disconnected type is locally profinite? Let $G$ be a Hausdorff topological group in which every point has a neighborhood basis of open compact neighborhoods.  Let's call this a group of totally disconnected (td)-type.
On the other hand, we have a notion of a locally profinite group, a Hausdorff topological group which has a neighborhood basis of the identity consisting of open compact subgroups.  A locally profinite group is obviously of td-tytpe.  
Is there an example of a group of td-type which is not locally profinite?  

I think $\mathbb Z$ would be a counter example : 
$\mathbb Z$ is discrete thus each point $n$ has a basis of compact open neighborhood given by the compact open set $\{n\}$. 
Howewer, subgroups of $\mathbb Z$ are of the form $n\mathbb Z$.
