I am looking for an example of an unbounded sequence $(x^{k})\_{k\in {\mathbb N}}$ such that $\|x^{k+1}-x^{k}\|\to 0$ and $(x^{k})\_{k\in {\mathbb N}}$ has at least one isolated cluster point x but the sequence is not convergent. Can this happen?

Let me clarify what seems to go wrong when trying to prove that $(x^k)\_{k\in{\mathbb N}}$ is convergent through contradiction. Lets assume the contrary that $x^k$ is not convergent. Then there exists $\epsilon>0$ and a set of infinite subindices $n\_k$ such that $\|x^{n\_k} - x\| > \epsilon>0$. Moreover, we know that there exists another set of indices $p\_l$ such that $\|x^{p\_l}-x\|\to 0$ as $l\to \infty$ (since $x$ is a cluster point).
Let $t(n\_k)$ denote the smallest index larger than $n\_k$ such that $t(n\_k)= p\_l$ for some $l$. Then using the triangular inequality
\begin{equation}
\epsilon < \|x^{n\_k}-x\| \leq \underbrace{\|x^{t(n\_k)}-x\|}\_{\to 0} + \underbrace{{\color{red}\sum\_{i={n\_k}}^{t(n\_k)-1}} \underbrace{\|x^{i+1}-x^i\|}\_{\to 0}}\_{\nrightarrow 0}
 \end{equation}
The issue is the last term where even though individual terms $\|x^{i+1}-x^i\|$ converge to $0$, it is possible that the distant between $t(n\_k)$ and $n\_k$ migh keep growing as $k\to \infty$ preventing the sum from converging to zero.

Did I miss something? Could you think of an example?

Thank you.