You made me doubt myself for a moment, but:

http://planetmath.org/compactnessofclosedunitballinnormedspaces

[cool math]Pretty sure the same is true of any metric space: cover the ball with open neighbourhoods of the endpoints of a basis of unit vectors where each such neighbourhood covers only one such endpoint, as well as any open sets that you need to cover the rest of the ball but that do not cover those endpoints. If you have an infinite number of dimensions then you won't be able to refine the cover to a finite one because each endpoint is covered by only one open set.[/cool math]

Thanks for an interesting discussion. We should probably continue it somewhere else if you want to, given the direction this is taking...