A time-nonlinear stage of the collapse of islands in freely suspended smectic nanofilms is observed and investigated. Islands thicker than a nanofilm are prepared and studied, which are unstable inside the dislocation loops, since they increase the energy of the film. Such instability leads to the decrease in the size of islands and is terminated by their collapse. The time dependence of the size of islands is measured experimentally. It is shown that the found dependence is in agreement with the theory of the dynamics of dislocation loops in smectic films developed earlier with allowance for the dissipation of energy in the film and in the meniscus. A nontrivial dynamic coupling between islands in a film resembling Ostwald ripening is also found, though the nonequilibrium kinetics of unstable islands, at which the hydrodynamic flow through a film leads to the decrease in sizes of one island and the increase in those of the other, rather than of the growth of the nucleation centers in the thermodynamically stable phase from the metastable state of the system (described by the Lifshitz-Slezov theory in films), is studied in our experiments.