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 ~~ not ~~ closed.

~~My~~ 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 $\|f(x)-f(y)\| < \|x-y\|$. )