Actually, it's not hard to construct such a solution.  Ferguson (Mathematical Statistics, 1967, pg. 83) gives an example where $r(\pi,\delta)$ is the probability that independent draws $x \sim \pi$ and $y\sim\delta$ are such that $ x > y$.  Then every $\delta$ is minimax, but $\inf_{\delta} \sup_{\pi} r(\pi,\delta) = 0$.

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.