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 $|u_n|\ge 1/2$.

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 $|u_n|\ge 1/2$. (Reason: if $|u_n|\ge1/2$ then $u_ {n+1}$ lies in the segment of endpoints $u _ n$ and $f(u_n)=e^{i\pi/4}\frac{u _ n}{|u _ n|}$, a segment that is disjoint from the disk $B(0,1/2)$, by elementary geometry considerations).

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

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 $|u_n|\ge 1/2$.