Let $F$ be a map on $\mathbb{R}^n$ and consider the dynamical system induced by $F$, i.e., all the orbits $x^k=F^k(x_0)$ with $x_0 \in \mathbb{R}^n$. Suppose that $x^\star$ is an asymptotically stable fixed point of $F$. Assume that $F$ is locally Lipschitz. It is well known (Converse Lyapunov Theorem) that there exists a Locally Lipschitz Lyapunov function $V$ that "certifies" the asymptotic stability of $x^\star$.
My question is: Is there a map $F$ together with a fixed point $x^\star$ that is asymptotically stable for which no norm is a Lyapunov function that certifies the asymptotic stability?
By the way: of course that the example will have to be in the asymptotically stable but not exponentially stable case.