The situation described is impossible!! With the same center and the same metric, the ball with smaller radius is contained always in the ball of bigger radius! I suppose one of two, the metric or the center has to be different. If it is the metric, just consider the discrete case. If it is the center, unfortunately, I don't have anything immediate in mind. I can't think a little about it, but it sounds strange to me because I am used to Banach or Frechet spaces, with each point being an accumulation one.