We observe a dependence of the damping of a confined mode of precessing ferromagnetic magnetization on the size of the mode. The micron-scale mode is created within an extended, unpatterned yttrium iron garnet film by means of the intense local dipolar field of a micromagnetic tip. We find that the damping of the confined mode scales like the surface-to-volume ratio of the mode, indicating an interfacial damping effect (similar to spin pumping) due to the transfer of angular momentum from the confined mode to the spin sink of ferromagnetic material in the surrounding film. Though unexpected for insulating systems, the measured intralayer spin-mixing conductance g_↑↓=5.3×10(19) m(-2) demonstrates efficient intralayer angular momentum transfer.