Abstract The decomposition of oceanic flow into its geostrophically balanced and unbalanced motions carries theoretical and practical significance for the oceanographic community. These two motions have distinct dynamical characteristics and affect the transport of tracers differently from one another. The launch of the Surface Water and Ocean Topography (SWOT) satellite provides a prime opportunity to diagnose the surface balanced and unbalanced motions on a global scale at an unprecedented spatial resolution. Here, we apply dynamic‐mode decomposition (DMD), a linear‐algebraic data‐driven method, to tidally‐forced idealized and realistic numerical simulations at submesoscale‐permitting resolution and one‐day‐repeat SWOT observations of sea‐surface height (SSH) in the Gulf Stream downstream of Cape Hatteras, a region commonly referred to as the separated Gulf Stream. DMD is able to separate out the spatial modes associated with sub‐inertial periods from super‐inertial periods. The sub‐inertial modes of DMD can be used to extract geostrophically balanced motions from SSH fields, which have an imprint of internal gravity waves, so long as the data extends long enough in time. We utilize the statistical relation between relative vorticity and strain rate as the metric to gauge the extraction of geostrophy.