The Maximum Modulus Principle is False for Banach space valued functions

False Theorem: Let $X$ be a complex Banach space, $U\subset \mathbb{C}$ open and connected and $f:U\to X$ holomorphic. If $\Vert \cdot \Vert \circ f$ has a global maximum, then $f$ is constant.

False Proposition: In the same situation as above: If $\Vert \cdot \Vert \circ f$ is constant, then $f$ is constant.

Counter example (to both): Let $X:=({\mathbb{C}}^2,\Vert \cdot {\Vert }_{\mathrm{\infty }})$ (maximum norm). Let $U=\{z\in \mathbb{C}:|z|<1\}$ and define $f:U\to X$ by $f(z):=(1,z)$. Then $f$ is holomorphic, since both component functions are. Furthermore for all $z\in \mathbb{C}:\Vert f(z){\Vert }_{\mathrm{\infty }}=max\{|z|,1\}=1$. Therefore $\Vert \cdot \Vert \circ f$ is constant. However $f$ is not constant.