As in the title, I am looking for examples of locally profinite groups (i.e. locally compact, totally disconnected groups) such that every neighbourhood of identity contains a compact, open and normal subgroup - for convenience let's call it property P. Such a property is true for profinite groups (i.e. compact and totally disconnected), so I am looking for other examples. In fact, the further from compactness the better.
I learnt that a theorem of Dantzig asserts that any locally profinite group has the property that every neighbourhood of identity contains a compact, open subgroup. However, there is no guarantee that it will be normal, which I care about.
So my question is, are there many non-compact examples of locally profinite groups with property P (if any)? Ideally, I'd love to see an example of a non-$\sigma$-compact, locally profinite group with property P. Thank you in advance for any suggestions.
Trivial examples of pro-discrete locally compact groups are direct products $K\times D$ with $K$ profinite and $D$ discrete. Thus you can get it non-compact, and non-$\sigma$-compact if you like. Some (not all) semidirect products $K\rtimes D$ work as well.