TITLE: How to define Continuous $f: B\rightarrow B$ without fixed points?
QUESTION [8 upvotes]: Let $X$ be a infinite dimensional Banach space and $B=\{x\in X:\ \|x\|\leq 1\}$.
How to define a continuous $f:B\rightarrow B$ without fixed points?
Edit 1: I have changed the question a litlle, because if $f$ is linear then $f(0)=0$.

REPLY [8 votes]: This paper shows that in an  arbitrary normed, infinite dimensional space, there is a continuous map on its unit ball (into its unit ball) that is fixed-point free with even stronger properties. 
According to the authors, the fact that every infinite dimensional normed space admits a fixed-point free, continuous map on its unit ball was first proved, using the Axiom of Choice, in Theorem 6.3 of [1], below, and was proven later  using "a more constructive proof", in [2], below. 
Of course, for specific spaces, it is not too hard to find fixed-free maps. For example, if $X=c_0$, then the map from $B_X$ to $B_X$ that sends the element $x=(x_1, x_2,\ldots)$ to $(1-\Vert x\Vert, x_1,x_2,\ldots)$ is continuous (in fact, non-expansive) and fixed-point free.  An example on the space $\ell_2$ is given in Patrick Da Silva's answer; and this can be generalized to examples for $\ell_p$ spaces as suggested by Tomás in the comments to your post.

[1] J. Dugundji, An extension of Tietze's theorem, Pacific Journal of math., I (1951), 353-367 
[2]  V. L. Klee, Jr., Some topological properties of convex sets, Trans. Amer. Math. Soc., 78, (1955), 30-45.