In the present article the author studies holomorphic functions on a certain class of pseudoconvex domains \(\Omega\) that are embedded as relatively compact domains in a holomorphic \(\mathbb{CP}^1\)-bundle over a compact Riemann surface. The boundary \(\partial\Omega\) of these domains \(\Omega\) is a real-analytic Levi-flat hypersurface, that is, it is foliated by Riemann surfaces. Such domains are of great interest in connection with the Levi problem on complex manifolds. From the general theory of holomorphic functions of several variables, one can easily see that \(\Omega\) admits plenty of non-constant holomorphic functions. In the present article, it is shown that \(\Omega\) also admits lots of non-constant holomorphic functions with slow growth, namely the weighted Bergman space \(A^2_\alpha(\Omega)\) of weight order \(\alpha >-1\) is infinite dimensional (Main theorem). These holomorphic functions are produced using an integral formula from holomorphic differentials on the Riemann surface. On the other hand, the Hardy space \(A^2_{-1}(\Omega)\) consists only of the constant functions. In particular, there are no bounded holomorphic functions on \(\Omega\) except the constant functions (Corollary 2).