Not an answer, but an intuition for an answer (suggesting that balls do not have exponentially small expansion).
Consider the three-regular infinite tree $T_3$. All balls $B_R$ of radius $R$ are isometric; they are all finite almost-binary trees. The number of vertices in $B_R$ is
$1 + 3(1 + 2 + 2^2 + \cdots + 2^{R-1}) = 3 \cdot 2^R - 2$
The number of leaves is $3 \cdot 2^{R-1}$. This is close enough to one-half of the vertices that we will ignore the error. The one-neighbourhood of the leaves, taken in $B_R$, of course has the same number of edges. So the expansion of the set of leaves is one. (If we instead take the other definition, and instead take the numerator to be the number of vertices in the one-neighbourhood of the leaves, then the expansion is $3/2$.)
The intuition is that this is the extreme case. If that is correct, then the expansion of $B_R$ is close to some constant, and thus not exponentially small in $R$.