Let $Y=(\omega+1)\times\omega$ with the product topology, let $p$ be a point not in $Y$, and let $X=\{p\}\cup Y$. Let $\mathscr{U}$ be a free (i.e., non-principal) ultrafilter on $\omega$, and let $\mathscr{F}$ be the cofinite filter on $\omega$. Basic open nbhds of $p$ are sets of the form
$$B(U,F)=\{p\}\cup\big(\{\omega\}\times U\big)\cup(\omega\times F)\;,$$
where $U\in\mathscr{U}$, and $U\subseteq F\in\mathscr{F}$. It’s not hard to check that $X$ is Hausdorff.
Let $D=(\omega\times\omega)\cup\{p\}$; clearly $D$ is dense in $X$. $\{D\cap B(F,F):F\in\mathscr{F}\}$ is a countable local base at $p$ in $D$, but $\chi(p,X)>\omega$, since $\mathscr{U}$ does not have a countable filterbase.