The linear polymer poly(p-phenylene terephthalamide), better known by its tradename Kevlar, is an icon of modern materials science due to its remarkable strength, stiffness, and environmental resistance. Here, we propose a new two-dimensional (2D) polymer, "graphamid", that closely resembles Kevlar in chemical structure, but is mechanically advantaged by virtue of its 2D structure. Using atomistic calculations, we show that graphamid comprises covalently-bonded sheets bridged by a high population of strong intermolecular hydrogen bonds. Molecular and micromechanical calculations predict that these strong intermolecular interactions allow stiff, high strength (6-8 GPa), and tough films from ensembles of finite graphamid molecules. In contrast, traditional 2D materials like graphene have weak intermolecular interactions, leading to ensembles of low strength (0.1-0.5 GPa) and brittle fracture behavior. These results suggest that hydrogen-bonded 2D polymers like graphamid would be transformative in enabling scalable, lightweight, high performance polymer films of unprecedented mechanical performance.