I was thinking about Brower's fixed point theorem. Every continuous function $f:D^n\rightarrow D^n$ has a fixed point where $D^n$ is the closed unit ball in $\mathbb{R}^n$. I started thinking of a generalization. If we consider $D^\infty\subseteq\ell^2$ where $D^\infty=\{(x_0,x_1,...)\vert\sum_{i=0}^\infty x_i^2\leq 1\}$. Must it be true that every continuous function $f:D^\infty\rightarrow D^\infty$ contains a fixed point. When looking for an answer, I discovered the Schauder fixed point theorem, but I am not sure if it is applicable here (I am primarily worried about the condition that requires $f(D^\infty)$ to be contained in a compact subset of $D^\infty$).
My intuition tells me that such a function with no fixed points exists; however, it isn't clear why. If someone knows of such a function with no fixed points, or of a proof of why no such function exists, I would be interested to learn.