I can imagine how one could pull it off.

1. 4 (or more) tension wires connected orthogonally ( like spokes in a wheel ) to a mounting plate that holds a water proof junction box. This will help keep the junction box and light from swinging all over the place and provide more support.
2. line voltage wire from the exterior outlet to the junction box. (Assuming line voltage lighting)
3. water proof light fixture hanging from the junction box.
4. Put a control ( switch) for the light circuit.

Some challenges to overcome

1. Setting up the tension wire system that will bear the weight of the light(s), withstand wind forces and stay up for years This would be necessary for string lights as well.
2. Water proofing the circuit.
3. Controlling the light.
4. Code compliance (assuming that is important).

A pergola provides a stable platform for attaching the necessary wiring and conduit.

For a movable (& not fixed) solution, you could use the base and arm for a cantilevered umbrella system to hang a pendant.

David, If I had places to connect four or more orthagonal lines, I would do that. But the trees aren't well-situated for that, and I'm not sure about installing fee-standing poles. They might be a little too intrusive, visually. (If we went with slimmer metal poles, rather than redwood posts, they might not stand out as much. I may need to investigate how strong metal poles would have to be, and how I could get them fabricated.) The cantilevered umbrella system is a clever idea, but I think that would be too big and intrusive for what we have in mind.

We have another outlet at the edge of the patio near the table. I suppose we could go with one of those under-umbrella lighting kits that would attach to the ribs of our existing market umbrella and run the cord from beneath the table to the outlet. We were hoping to have lighting that leaves the dining area open to the sky, but that may not be practical.