Most potential energies appear to have their basis in a field, but do all? I know gravitational energy has the form $mgh$, which has a position term $h$ but no velocity. More "internal" energies like elastic energies in a spring can find their root in atomic forces, such as electrostatic forces. Indeed, when we look at Lagrangian mechanics, we find that the Lagrangian form of Netwonian physics neatly splits the Lagrangian into $L=T-V$, where $T$ is the kinetic energy in the system (depending on velocity), and $V$ is the potential energy (depending only on position).
Is this always true? Based on current theories, do we always find potential energy takes a form based on a position in spatial coordinates, and nothing more? Or do we find exotic concepts of energy that show up in the extremes (such as the subatomic world) that are not captured in this field-centric way?