A separable, ultrametric space $X$ is given. Does that mean that one immediately gets a countable cover of $X$ consisting of open balls with radius $r>0$ by virtue of separability? (I mean to center a ball with radius $r>0$ at every point of the countable dense subset $U$.)
My question is this: I understand the statement for metric spaces. Does it also hold for ultrametric spaces? Do I have to show anything additionally?