The proximal map is defined in the sense of an operator as:
$\text{prox}_{\lambda f}(x) = \arg \min_y f(y) + \frac{1}{2\lambda}\|x-y\|^2$
I don't see why it is called a mapping. I suppose it is because $\arg \min$ might "return" not only one value but a few of them. What is an example of this situation?
What is an intuitive explanation why the proximal mapping is called a mapping? Can you give an example?
It's just because the word "mapping" is a synonym for "function". The proximal mapping is also called the "proximal operator". If $f$ is convex and lower semicontinuous then the $\arg \min$ set in question is guaranteed to be a singleton.