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Paul Broussous has indicated the way the Steinberg representation enters into the study of the group over a non-archimedean local field. I'll add some comments, in community-wiki format, to deal wit …

The whole subject of BN-pairs and buildings started with Tits, but the emphasis on local fields began with the work of Iwahori and Matsumoto in the mid-1960s. … Anyway, it's a large literature by now, including books on buildings, which somewhat obscures these special cases of interest. …

ADDED: To follow up Greg's comment, there have been other items in the Notices dealing directly with Bruhat-Tits buildings. An article by K.S. …

But the theory of buildings lies beyond this elementary axiomatic stage and is not directly relevant to the question asked here. … Maybe it's also worth mentioning that quite a few MO questions involve BN-pairs, also known as (B,N)-pairs or Tits systems, as well as buildings. So it may be worthwhile to browse through these. …

To replace my somewhat fuzzy comment, maybe I can formulate a skeptical semi-answer. At any rate your question probably doesn't have a clearcut answer unless you impose strong enough restrictive cond …

Though I'm not at all a specialist in buildings, I know some of the complicated history of the subject as it evolved into long and highly sophisticated papers by Bruhat-Tits and others. … Once you get beyond split groups and ordinary $p$-adic extensions of the rationals, a lot more machinery has to be developed in order to work effectively with buildings and subgroup actions on them. …

In that conventional setting, which is close to the geometry of buildings and apartments, the Weyl group and its length function play a vital role in talking about distances in the geometry, etc. … Naturally much of this shows up in his own Springer lecture notes on the "spherical" case as well as in later books on buildings. …