This study examines spreading of surface drifter pairs deployed as part of the CLIVAR Mode Water Dynamic Experiment (CLIMODE) project in the Gulf Stream region. The spreading is resolved at hourly resolution and quantified by relative dispersion and finite‐scale Lyapunov exponents. At scales from 1–3 km to 300–500 km, the dispersion follows Richardson's law, indicating stirring by eddies comparable in scale to the pair separation distance. At larger scales, the spreading becomes a random walk described by a constant diffusivity. The behavior from 1–3 km to the local deformation radius is inconsistent with the enstrophy cascade of 2‐D quasigeostrophic turbulence. To test various hypotheses for this result, drifter pair spreading is examined for pairs that were not launched together, pairs deployed in the eastern subtropical North Atlantic, and CLIMODE pairs subsampled to daily temporal resolution. Our results indicate the presence of significant energy at the submesoscale in the Gulf Stream region which flattens the wave number spectrum and dominates surface stirring at this scale range. Results in the less energetic subtropical eastern Atlantic are more equivocal.