Are there any Banach space $X$ with $\operatorname{dim}(X)=\infty$ satisfying $S_X=\lbrace x\in X| |x|=1\rbrace $ is covered by for some $B_1,B_2,\ldots,B_N$, where $B_N$ are balls in $X$ with $0\notin B_i$ for $i=1,\ldots, N$?
I can find some literatures concerning countably many ball cases (e.g. T. W. Koerner, J. Lond. Math. Soc. (1970) 643-646). Also, I can prove that there are no such Hilbert space $X$ (using, for example, orthonormal bases?)
But, I couldn't find any literature about finite ball case (which is my question). Is it trivial?