Surjective isometries on the positive parts of the unit spheres of some function spaces

We consider the space $C^1[0, 1]$ of continuously differentiable functions on the closed unit interval $[0, 1]$ and the space $\operatorname{Lip}[0, 1]$ of Lipschitz continuous functions on $[0, 1]$, equipped with the norms \begin{align*} \|f\|_{\sigma, p} = \begin{cases} \sqrt[p]{|f(0)|^p + \|f'\|_\infty^p} & (1 \le p < \infty), \\ \max\{\, |f(0)|, \|f'\|_\infty \,\} & (p = \infty). \end{cases} \end{align*} We show that every surjective isometry on the positive part of the unit sphere extends to a surjective complex-linear isometry on the entire space.

As a corollary, every such isometry also extends to an isometric order isomorphism on the real subspaces $C^1_{\mathbb{R}}[0, 1]$ and $\operatorname{Lip}_{\mathbb{R}}[0, 1]$.