For discrete groups, the Folner characterization of amenability says that for any finite subset $K\subseteq G$, and any $\varepsilon$, there exists a finite subset $F$ such that $|F\Delta KF|<\varepsilon |F|$. If moreover $G$ is countable, we have the stronger property of the existence of a Folner sequence, namely a sequence of finite subsets $(F_n)$ such that for any $\gamma \in G$, $|F_n \Delta \gamma F_n|/|F_n|\rightarrow 0$.
Are there uncountable discrete amenable groups that also have Folner sequences? Or does the the existence of a Folner sequence imply countability of the underlying group?
Trivially no/yes (to the first and second question, which are negation of each other): uncountable groups have no Følner sequences.
Indeed, let $\Gamma$ be an uncountable group and let $(F_n)$ be a sequence of nonempty finite subsets. Let $\Lambda$ be the subgroup generated by $\bigcup_nF_n$. Since $\Lambda$ is countable, there exists $\gamma\notin\Lambda$. So $\gamma F_n\cap F_n$ is empty for all $n$. In other words, $|F_n\Delta\gamma F_n|/|F_n|=2$ for all $n$. So $(F_n)$ is not a Følner sequence.