Actually, it's not hard to construct such a solution. Ferguson (Mathematical Statistics, 1967, pg. 83) gives an example where $\Delta, \Pi$ are the space of probability measures on $\mathbb{N}$ and $r(\pi,\delta)$ is the probability that independent draws $x \sim \pi$ and $y\sim\delta$ are such that $ x > y$. Then every element in $\Delta$ is minimax because $\sup_{\pi} r(\pi,\delta) = 1$ for all $\delta$, but $\sup_{\pi} \inf_{\delta} r(\pi, \delta) = 0$ (but this supremum is not achieved).
I see that it is also easy to produce a game where not every $\delta$ is minimax. I suppose a more interesting example would be one where the minimax was unique, yet the game had no value. I dot't yet see a way to do this. Perhaps this is ruled out logically, but I don't see it yet.