This fails for $n = 2$. For simplicity consider the unit square $C = [0,1]^2$ instead of $B$. Then define $f$ by first setting $f(x,y) = (x + 1/2, y)$ for $(x,y) \in C$ with $x \leq 1/2$ --- this shifts the left half of the square onto the right half, no fixed points there. Define $f(x,y) = (2x - 3/2, y)$ for $(x,y) \in C$ with $x \geq 3/4$ --- this takes the right one-fourth of $C$ onto the left half, again no fixed points. Then the middle strip $[1/2, 3/4] \times [0,1]$ can be stretched around over the top of the square to complete the definition of a continuous map. I have a hard time describing that last step without drawing a picture, is it clear?