We consider families of analytic area-preserving maps depending on two pa- rameters: the perturbation strength E and the characteristic exponent h of the origin. For E=0, these maps are integrable with a separatrix to the origin, whereas they asymptote to flows with homoclinic connections as h->0+. For fixed E!=0 and small h, we show that these connections break up. The area of the lobes of the resultant turnstile is given asymptotically by E exp(-Pi^2/h)Oª(h), where Oª(h) is an even Gevrey-1 function such that Oª(0)!=0 and the radius of convergence of its Borel transform is 2Pi^2. As E->0 the function Oª tends to an entire function Oº. This function Oº 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.