Here is the simplest example that I can think of, but there may be simpler examples.  Let $(E,0)$ be an elliptic curve over $\mathbb{C}$.  On the surface $E\times E$, form the Cartier divisor $$D = \underline{\Delta} - \underline{\{0\}\times E} - \underline{E\times\{0\} },$$ where $\Delta$ is the diagonal.  Denote by $\mathcal{L}$ the associated invertible sheaf, $\mathcal{O}_{E\times E}(D)$.  Now form the geometric vector bundle $\pi:V\to E\times E$ such that there is a universal $\mathcal{O}_V$-module homomorphism, $$s:\pi^*(\mathcal{L}\oplus \mathcal{L})\to \mathcal{O}_V,$$ i.e., $V$ is $\text{Spec}_{E\times E}\text{Sym}^\bullet(\mathcal{L}\oplus \mathcal{L})$.  The dualizing sheaf of $V$ is isomorphic to $\pi^*( \mathcal{L}^{\otimes 2} )$.  Denote by $U\subset V$ the open complement of the zero section of $\pi$.  There is a scaling action of $\mathbb{G}_m$ on $V$, compatible with the projection to $\pi$: just scale $s$ by the universal invertible global section $t$ of $\mathcal{O}_{\mathbb{G}_m}$.  Let $q\in \mathbb{C}^*$ be an element of complex modulus different from $1$.  Then the scaling of $U(\mathbb{C})$ by $q\in \mathbb{G}_m(\mathbb{C})$ is a free, proper, discontinuous action with respect to the Euclidean / analytic topology on $U(\mathbb{C})$.  Thus, there is a well-defined quotient in the category of complex analytic spaces, and this quotient is even a compact complex manifold with a projection to the underlying analytic space of $E\times E$, $$\rho:M\to (E\times E)^{\text{an}}.$$  In particular, there is a projection of $(E\times E)^{\text{an}}$ to $E^{\text{an}}$ by projection on the first factor, $$M \xrightarrow{\rho} (E\times E)^{\text{an}} \xrightarrow {\text{pr}_1} E^{\text{an}}.$$  The relative dualizing sheaf of $\text{pr}_1\circ \rho$ is still $\rho^*(\mathcal{L}^{\otimes 2})$.  Of course the restriction of $\mathcal{L}^{\otimes 2}$ to the fiber over $0\in E^\text{an}$ is the structure sheaf, which has nonzero global sections.  But for $p\in E^{\text{an}}$ not a $2$-torsion point, the restriction of $\mathcal{L}^{\otimes 2}$ to the fiber over $p$ has only the zero section.