  In this paper we study the global geometry of the Kobayashi metric on domains
in complex Euclidean space. We are particularly interested in developing
necessary and sufficient conditions for the Kobayashi metric to be Gromov
hyperbolic. For general domains, it has been suggested that a non-trivial
complex affine disk in the boundary is an obstruction to Gromov hyperbolicity.
This is known to be the case when the set in question is convex. In this paper
we first extend this result to $\mathbb{C}$-convex sets with $C^1$-smooth
boundary. We will then show that some boundary regularity is necessary by
producing in any dimension examples of open bounded $\mathbb{C}$-convex sets
where the Kobayashi metric is Gromov hyperbolic but whose boundary contains a
complex affine ball of complex codimension one.
