Let $X$ be a Hadamard space. Any two points $x$ and $y$ of $X$ have a unique midpoint $m = m(x,y)$.

Given $x$ and $y$ any two points of $X$, is it always possible to find an affine function $f : X \rightarrow \mathbb{R}$ with $f(x) \neq f(y)$ ? (where by "affine" I mean $\forall x',y' : f(m(x',y')) = \frac{f(x') + f(y')}{2})$ ?