Question on contraction map into the euclidean ball

Question

How to define a map $f$ from $\mathbb R^n$ to the euclidean unit ball in such a way that $f$ is a contraction? I was thinking about the map that sends $x\to\frac{x}{\|x\|}$, where $\|x\|$ denotes the euclidean norm of $x$, but the image of this map is the sphere $S^{n-1}$, not the whole ball.