Let $K$ be an algebraically closed valued field which is $C$-minimal, as defined, for example, in this article. Examples include pure algebraically closed valued fields, as well as Lipschitz and Robinson's expansions by rigid subanalytic sets (Lipshitz. Rigid subanalytic sets. Amer. J. Math., 115(1):77-108, 1993)
Is it true then that the structure induced on the value group of $K$ is that of a vector space (over a field that depends on $K$)? This would mean, in particular, that definable sets are Boolean combinations of sets defined by systems of affine inequalities (i.e. polyhedra).