I am getting the “Accessory Not Supported” message on my iPad Pro, despite the fact that there is no accessory connected to it.

Here's the support information covering that error:

If you see an ‘Accessory may not be supported’ alert - Apple Support - https://support.apple.com/HT208277

but since you have nothing connected I think you'll end up having to contact Apple as the situation described halfway down recommends.

Are you using an Apple Smart (folio) keyboard?

This alert is typically characteristic of a faulty folio keyboard. The electrical connections that pass though the flexible fabric hinges typically fracture - this usually resulting in errors and/or the on-screen notification that you see.

The Smart folio keyboard is not repairable. The only remedy is replacement of the keyboard.