By Banach fixed point theorem, if a metric on a metric space $X$ is such that $d(f(x),f(y))\leq K d(x,y)$ for $K\in (0,1)$ then $f$ has one unique fixed point.
Is there an example where $d(f(x),f(y))\leq K d(x,y)$ does not have a fixed point if $K=1$?
What if $X$ is a compact space?