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A counterexample in dimension $2$: take $K$ the unit closed disk of $\mathbb{C}$, and $f:K\to K$ the map $$f(z)=e^{i\pi/4}\frac{z}{|z|}\min(2|z|,1)$$ for $z\neq0$, and $f(0)=0$, which is the only fixed point of $f$. Then, starting by $u_0\in K$ with $|u _ 0|\ge1/2$ produces a sequence with $|u_n|\ge 1/2$.