We report on a high-spatial-resolution survey for binary stars in the periphery of the Orion Nebula Cluster, at 5 - 15 arcmin (0.65 - 2 pc) from the cluster center. We observed 228 stars with adaptive optics systems, in order to find companions at separations of 0.13 - 1.12 arcsec (60 - 500 AU), and detected 13 new binaries. Combined with the results of Petr (1998), we have a sample of 275 objects, about half of which have masses from the literature and high probabilities to be cluster members. We used an improved method to derive the completeness limits of the observations, which takes into account the elongated point spread function of stars at relatively large distances from the adaptive optics guide star. The multiplicity of stars with masses >2 Msun is found to be significantly larger than that of low-mass stars. The companion star frequency of low-mass stars is comparable to that of main-sequence M-dwarfs, less than half that of solar-type main-sequence stars, and 3.5 to 5 times lower than in the Taurus-Auriga and Scorpius-Centaurus star-forming regions. We find the binary frequency of low-mass stars in the periphery of the cluster to be the same or only slightly higher than for stars in the cluster core (<3 arcmin from θ1C Ori). This is in contrast to the prediction of the theory that the low binary frequency in the cluster is caused by the disruption of binaries due to dynamical interactions. There are two ways out of this dilemma: Either the initial binary frequency in the Orion Nebula Cluster was lower than in Taurus-Auriga, or the Orion Nebula Cluster was originally much denser and dynamically more active.