Norm-Additive Maps on the Positive Unit Sphere of $C^1([0,1])$

Let $C^1(I)$ denote the space of all continuously differentiable real-valued functions on the closed unit interval $I=[0,1]$. For $1\le p\le\infty$, we equip $C^1(I)$ with the norm \[ \Vert{f}\Vert_{(p)} := \begin{cases} \left(|f(0)|^p+\Vert{f'}\Vert_{I}^p\right)^{1/p}, & 1\le p<\infty, \\ \max\{|f(