Is there any example of a symplectic manifold that has infinite volume, but which cannot be packed by an infinite number of symplectic balls of a fixed radius $r$, for any $r > 0$ ?

Note that on such a manifold we would be able to prove the "Poincaré recurrence" of every symplectomorphism.

Added. The problem I see with using Gromov's non-squeezing theorem (or any other argument based on capacities) as in the comment below is the following: a priori at least, part of the infinitely many small balls---the symplectic "ribs" of relatively large capacity---can perhaps be put into a fixed large symplectic ball inside the manifold, while putting the "ribless parts" of these balls responsible for their volumes can be put elsewhere. I can't see how to control this.