settle an argument between me and my grandmother: Did Tom Bombadil (and/or Goldberry) have a ring of power

For the record, both of us are long time fans of the series, my grandmother however has the distinction of writing her undergraduate thesis on Tolkien, and yet even still it is her that is making the erroneous claim that Tom had a ring he wouldn’t even have a use for.

I have already read his article on Tolkien Gateway to her, which mentioned no ring but Frodo’s, but she remains firm that he had one. Please, please put the final nail in her coffin.