Almost isometries of balls

Let $f$ be a bi-Lipschitz mapping of the Euclidean ball $B_{\mathbb{R}^n}$ into $\ell_2$ with both Lipschitz constants close to one. We investigate the shape of $f(B_{\mathbb{R}^n})$. We give examples of such a mapping $f$, which has the Lipschitz constants arbitrarily close to one and at the same time has in the supremum norm the distance at least one from every isometry of $\mathbb{R}^n$.