On 2-local nonlinear surjective isometries on normed spaces and C$^*$-algebras

We prove that, if the closed unit ball of a normed space $X$ has sufficiently many extreme points, then every mapping $\Phi$ from $X$ into itself with the following property is affine: For any pair of points in $X$, there exists a (not necessarily linear) surjective isometry on $X$ that coincides with $\Phi$ at the two points. We also consider surjectivity of such a mapping in some special cases including C$^*$-algebras.