Normed structures, and notably normed groups, valuated rings, normed (vector) spaces (over a valuated field), and normed algebras among the others, are familiarly considered as sort of special metric spaces. Thus, according to the common standpoint, one would say that "metrics are on a higher step than norms on the hierarchy ladder of abstraction" (basically for the fact that we have a way to equip them with a "seemingly natural" metric but no "seemingly natural" way to think of an arbitrary metric space as a normed structure), and I was actually taught to think of it in this mood. But now, I'm really wondering if there exists previous work providing a (completely) different outlook, or at least suggesting that the "truth" may be somewhere else in the other view. Of course, I've already looked through Lawvere's well-known paper on metric spaces and generalized logic, but no hints seem to be there to the picture that I'm trying to draw. Then, my question is:
Are you aware of any previous literature pursuing the idea that metrics can be ultimately regarded as a special kind of "norms", to the degree that metric spaces can be recovered as an instance of abstract normed structures of some appropriate type (where the word type has, say, the formal meaning of model theory)?
I've no clue how to tag this question. I was thinking of "total temporary insanity", but I couldn't find it in the list.
P.S.: this is an edit following up the comments below (where the interested reader can find some references to the ubiquitous presence of norms beyond the scope of the theory of Banach spaces).