I think that you assumed that $a_i\ne 0$ for all $i$, otherwise an easy "no" answer is given by the vector $(1,0,\dots,0)$ in the Euclidean space.

Assuming that this correction was made, consider the following space: $\ell_\infty^2\oplus_\infty\ell_2^{n-2}$. This is the space of sequences of length $n$ such that the norm of the sequence is the maximum of the $\ell_\infty$-norm of the first two elements and the Euclidean norm of the remaining elements. It is easy to see that for any of the extreme points in the dual ball one of the first two coordinates is $0$.