Is there a locally countable and weakly Lindelöf space which is not ccc?

A space $X$ is locally countable if for each point $x\in X$ there is an open neighbourhood $O_x$ of $x$ such that $|O_x| < \omega_1$;

Here ccc denotes the countable chain condition;

A space $X$ is called weakly Lindeöf if for any open cover $\mathcal U$ of $X$ there is a counable subset $\mathcal V \subset \mathcal U$ such that $\bigcup \mathcal V$ is dense in $X$.

Thank you.