We report on a qualitative change of the character of interacting magnon excitations in a ferromagnet due to finite size. Within an improved continuum approach excitation modes are found analytically for a two-dimensional system. Several non-totally symmetric modes are obtained which were previously unknown. In contrast to excitations in one dimension some of the low energy modes approach the macroscopic limit extremely slowly. These modes occupy a substantial part of the Brillouin zone and can therefore be responsible for the logarithmic size dependence of various thermodynamic quantities.