Do Gap-CDH groups exist?
The closest that I can think of is a "symmetric bilinear group" (a.k.a. Type-I pairing group) that was popular when bilinear groups were first introduced. This is actually a pair of groups $(G, G_T)$ together with an efficient non-degenerate bilinear map $\otimes: G \times G \to G_T$. Obviously DDH is easy in $G$, since one can on input $(g, U, V, W) \in (G_1)^3$ test whether $g \otimes W = U \otimes V$. But the pairing will not necessarily help you compute CDH in the source group, since the map $(\cdot \otimes g)$ is not meant to be easy to invert.

These kinds of groups are constructed based on various types of elliptic curve, my intuition is that a pairing "out of" a curve group won't help you compute new elements in the group itself.

This kind of group appeared in Boneh, Boyen and Shacham (eprint 2004/174) where they used the Decision Linear assumption instead of DDH, since DLIN can be hard even in the presence of a pairing.