I am interested in constructing the following "counter-example" to the Banach's fixed point theorem.

Let $K=$ {$ g\in L_1: \|g\|=1, g(\cdot)\ge0 $}.
 Clearly, $K$ is not a compact and $K$ is <s> not </s>  closed.

 <s>My</s> A question is: is it possible to construct a [*edit*] *nonexpansive* mapping $f: K\to K$  with no fixed point? (   i.e. a mapping $f$ such that [*edit*] for all $x\neq y\in K$ one has <s>$\|f(x)-f(y)\| <  \|x-y\|$. )