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Using the first epoch of four-band NIRCam observations obtained by the James Webb Space Telescope (JWST) Prime Extragalactic Areas for Reionization and Lensing Science Program in the Spitzer IRAC Dark Field, we search for F150W and F200W dropouts. In 14.2 arcmin^2, we have found eight F150W dropouts and eight F200W dropouts, all brighter than 27.5 mag (the brightest being ~24 mag) in the band to the red side of the break. As they are detected in multiple bands, these must be real objects. Their nature, however, is unclear, and characterizing their properties is important for realizing the full potential of JWST. If the observed color decrements are due to the Lyman break, these objects should be at z >~ 11.7 and z >~ 15.4, respectively. The color diagnostics show that at least four F150W dropouts are far away from the usual contaminators encountered in dropout searches (red galaxies at much lower redshifts or brown dwarf stars). While the diagnostics of the F200W dropouts are less certain due to the limited number of passbands, at least one of them is likely not a known type of contaminant, and the rest are consistent with either high-redshift galaxies with evolved stellar populations or old galaxies at z ~ 3 to 8. If a significant fraction of our dropouts are indeed at z ~ 12, we have to face the severe problem of explaining their high luminosities and number densities. Spectroscopic identifications of such objects are urgently needed.