Wikipedia's article on ultrametric spaces seems to suggest that an ultrametic space can also be a normed vector space.
It seems to be impossible for an ultrametric to be induced by a vector space norm, because if $v$ is a nonzero vector, then $$d(0,2v)=\|2v\|=2\|v\|>\|v\|=\max(d(0,v),d(v,2v)) $$ contradicting $d$ being an ultrametric.
So what's going on there? Is there a concept of "normed vector space" that dispenses with the requirement that scalar multiplication scales the norm accordingly? (That would be more of a "normed abelian group", then).