In mammalian neocortex, the orderly arrangement of columns of neurons is thought to be a fundamental organizing principle. In primary visual cortex (V1), neurons respond preferentially to bars of a particular orientation, and, in many mammals, these orientation-selective cells are arranged in a semiregular, smoothly varying map across the cortical surface. Curiously, orientation maps have not been found in rodents or lagomorphs. To explore whether this lack of organization in previously studied rodents could be attributable to low visual acuity, poorly differentiated visual brain areas, or small absolute V1 size, we examined V1 organization of a larger, highly visual rodent, the gray squirrel. Using intrinsic signal optical imaging and single-cell recordings, we found no evidence of an orientation map, suggesting that formation of orientation maps depends on mechanisms not found in rodents. We did find robust orientation tuning of single cells, and this tuning was invariant to stimulus contrast. Therefore, it seems unlikely that orientation maps are important for orientation tuning or its contrast invariance in V1. In vertical electrode penetrations, we found little evidence for columnar organization of orientation-selective neurons and little evidence for local anisotropy of orientation preferences. We conclude that an orderly and columnar arrangement of functional response properties is not a universal characteristic of cortical architecture.