We consider a pointlike vortex in a layered superconductor with linear defects in the superconducting layers. We treat these defects as Josephson junctions with high critical current density. We consider the electrodynamics of these junctions within the framework of nonlocal Josephson electrodynamics. We show that Josephson current through a linear defect in a superconducting layer results in a pointlike vortex with a superconducting core residing in this layer (Josephson pancake). We find the mobility of a Josephson pancake. We consider a small amplitude wave in a Josephson junction with nonlocal electrodynamics. We treat a bending wave for an infinite stack of Josephson pancakes. We find the dispersion relation for these waves. We show that their velocities tend to a certain finite limit when the wavelength tends to infinity.