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 $\|\cdot \| \circ f$ has a global maximum, then $f$ is constant.

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

Counter example (to both): Let $X := (\mathbb{C}^2, \| \cdot \|_\infty)$ (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 $z \in \mathbb{C} : \| f (z) \|_\infty = \max \{ |z| , 1\} = 1$. Therefore $\| \cdot \| \circ f$ is constant. However $f$ is not constant.