Abstract. We consider families of analytic area-preserving maps depending on two parameters: the perturbation strength $\varepsilon$ and the characteristic exponent $h$ of the origin. For $\varepsilon=0$, these maps are integrable with a separatrix to the origin, whereas they asymptote to flows with homoclinic connections as $h\rightarrow 0^{+}$. For fixed $\varepsilon\neq 0$ and small $h$, we show that these connections break up. The area of the lobes of the resultant turnstile is given asymptotically by $\varepsilon \exp(-\pi^{2}/h)\Theta^{\varepsilon} (h)$, where $\Theta^{\varepsilon} (h)$ is an even Gevrey-1 function such that $\Theta^{\varepsilon} (0)\neq 0$ and the radius of convergence of its Borel transform is $2\pi^{2}$. As $\varepsilon\rightarrow 0$, the function $\Theta^{\varepsilon}$ tends to an entire function $\Theta^{0}$. This function $\Theta^{0}$ agrees with the one provided by the Melnikov theory, which cannot be applied directly, due to the exponentially small size of the lobe area with respect to $h$. These results are supported by detailed numerical computations; we use an expensive multiple-precision arithmetic and expand the local invariant curves up to very high order.