Let $B$ be the closed unit ball in $\mathbb R^3$ and $f: B\to B$ continuous, such that $f\circ f$ is the identity (i.e., $f\circ f=\mathbb 1_B$) and $f$ restricted on $\partial B$ is also the identity (i.e., $f|_{\partial B}=\mathbb 1_{\partial B}$). Does it imply that $f$ is the identity on $B$?

EDIT. If $B$ is instead the closed unit ball in $\mathbb R^2$, then the answer is positive. (Am. Math. Monthly 1994.)