I have a counterexample in $\mathbb R^2$.
Here's how it goes.<br>

Pick two numbers $n$ and $N$, with $N>>n>>1$.<br>
The collection {$v_i$} consists of:

- $N(n-1)$ times the vector $(\frac{n-2}{n},\frac{1}{N(2n-3)})$

- $N(n-2)$ times the vector $(-\frac{n-1}{n},\frac{1}{N(2n-3)})$

- The vector $(0,-1)$ once.

The smallest ball into which those vectors can be fit back-to-back has diameter $\sqrt{5}-\varepsilon$.