Question. Is there any example of a symplectic manifold $(M,\omega)$ 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:

Given a symplectomorphism $T: M \rightarrow M$ and an open set $U \subset M$ there exists an integer number $k > 0$ such that $T^k(U)$ intersects $U$.

Added and edited: It seems to me that the examples suggested in the comments assume the validity of the following (to my knowledge unproved) statement:

Let $(M,\omega)$ be the symplectic manifold obtained by taking $B(R)$, the symplectic ball of radius $R$ and dimension 2n, and attaching a long thin cylinder with very large volume and very small capacity. Let $r < R$ and consider the ball $B(r)$. There exists constant $c > 0$ such that any symplectic embedding of $B(r)$ into $M$ intersects the ball $B(R) \subset M$ in a set whose volume is at least $c$.

Note that this is not quite the intuition of non-squeezing theorem which would just say that you cannot embedd $B(r)$ into $M$ if $r > R$.