Sheep are gregarious animals that can form close inter-individual relationships. Social bonds are affected by environmental and within-animal factors, but there is limited understanding of individual factors that influence social relationships, in part due to the challenges of accurately measuring individual social proximity within a group. This study used geospatially precise RTK-GNSS localization devices (Real-Time-Kinematic Global Navigation Satellite System) to evaluate social proximity between individuals with different personality traits in small sheep flocks. First, unsupervised clustering algorithms were used to categorize 78 individual Merino sheep (from a research flock of 108 grazed together for 11 months prior) based on their behavior measured in four behavioral tests: arena, novel object, flight speed, and isolation box. Sixty-five sheep were then categorized into five cohorts of 13 individuals each (clustered into three bold, six bold/fearful, and four fearful animals per cohort). Bold/fearful animals were bold toward a novel object but fearful of a human. Sheep within a single cohort were released into a 0.70 Ha test paddock with RTK-GNSS devices attached to their backs for 3 days to monitor relative positions. All cohorts were tested sequentially over a 5-week period. Social network analyses of GNSS locational data with a distance threshold of up to 30 m for all cohorts together showed no significant effect of personality traits (P = 0.11). There were, however, some significant but inconsistent differences in distances based on personality traits within separate cohorts [i.e., cohorts 1, 3, 4 (P ≤ 0.02), cohorts 2, 5 (P ≥ 0.44)]. When GNSS locational data were analyzed with a distance threshold of up to 2 m (day 2 excluded from all cohorts due to missing values), there was a significant effect of personality traits (P = 0.01) across all cohorts combined. The bold animals showed greater distances from each other relative to their distances from fearful or bold/fearful individuals. Additionally, regardless of personality traits, the average distance between animals decreased 2.2 m over the 3 test days. Social interactions in sheep flocks may be influenced by personality traits of fearfulness and boldness, but further research with larger numbers of sheep across varying environments is needed.