There is such a space. For any $\mathscr{U}\in\beta\omega\setminus\omega$ we can start with the subspace $\omega \cup \{\mathscr{U}\}$ of $\beta\omega$ and ‘fatten up’ each isolated point to a copy of the rationals.
Let $\mathscr{U}$ be a free ultrafilter on $\omega$. Let $p$ be a point not in $\omega\times \mathbb{Q}$, and let $X = (\omega\times \mathbb{Q})\cup \{p\}$. We topologize $X$ as follows. For each $q\in\mathbb{Q}$ let $\mathscr{B}(q)$ be the set of clopen nbhds of $q$ in the usual topology on $\mathbb{Q}$. For $\langle n,q \rangle \in \omega\times \mathbb{Q}$ let $\mathscr{B}(n,q) = \{\{n\}\times B:B \in \mathscr{B}(q)\}$ be a local base at $\langle n,q\rangle$. Finally, take $\mathscr{B}(p) = \{U\times \mathbb{Q}:U\in\mathscr{U}\}$ as a local base at $p$. Let $\mathscr{B} = \mathscr{B}(p)\cup \bigcup_{\langle n,q\rangle\in \omega\times \mathbb{Q}}\mathscr{B}(n,q)\;;$ then $\mathscr{B}$ is a base for a topology $\mathscr{T}$ on $X$, and it’s easy to check that $\langle X,\mathscr{T}\rangle$ has no isolated points, is not first countable at $p$, and is regular. Indeed, the members of $\mathscr{B}$ are clopen in $\langle X,\mathscr{T}\rangle$, so $\langle X,\mathscr{T}\rangle$ is zero-dimensional and hence completely regular.
The same idea of ‘fattening up’ isolated points to isolated copies of $\mathbb{Q}$ can be applied to the Arens-Fort space. Start with $Y={\langle 0,0\rangle}\cup (\mathbb{Z}^+\times\mathbb{Z}^+$), where each point of $\mathbb{Z}^+\times\mathbb{Z}^+$ is isolated, and a set $V$ containing $\langle 0,0\rangle$ is open iff $\{m\in\mathbb{Z}^+:V\setminus(\{m\}\times\mathbb{Z}^+)\text{ is infinite}\}$ is finite (i.e., $V$ contains all but finitely many points of all but finitely many ‘columns’ of $\mathbb{Z}^+\times\mathbb{Z}^+$).
To get the desired space $X$, first replace each $\langle m,n\rangle \in \mathbb{Z}^+\times\mathbb{Z}^+$ by a copy, $Q(m,n)$, of $\mathbb{Q}$ with its usual topology. If $V$ is an open nbhd of $\langle 0,0\rangle$ in $Y$, let $V^* = \{\langle 0,0\rangle\}\cup \bigcup_{\langle m,n\rangle\in V\setminus\{\langle 0,0\rangle\}} Q(m,n),$ and take the family of such sets $V^*$ as a local base at $\langle 0,0\rangle$. The resulting space is countable, has no isolated points, is not first countable at $\langle 0,0\rangle$, and has a clopen base.