At first the answer seems simple, no, but the more I think about it the more it seems this might be possible.
For the sake of the question don’t think about the probability of this actually happening, but it must be able to occur naturally. Think of a very small star like a white dwarf or brown dwarf orbiting a super massive terrestrial or gas planet.
Bonus question: if such an anomaly could occur, despite the odds how would it come to be?
Mass is what makes something a star. Mass is also what warps spacetime to create gravity.
Imagine two round objects in space. If one of them has enough mass to be a star, and the other one has too little mass to be a star, then the bigger one will have a much deeper gravity well and the smaller thing will orbit around it.
Crudely speaking, small things always orbit big things.
More precisely, both objects will orbit their combined center of mass, which in astronomy is known as their barycenter.
The position of the barycenter is based on the relative masses of the two objects. If the star is big and the planet is small, their barycenter can be inside the body of the star (but not at the star's exact center). If the star is small-for-a-star and the planet is large-for-a-planet, the barycenter can be in open space between the two. But even then, it will be closer to the star than the planet.
