No, because a star is much more massive than a planet. In a two-body system, the smaller body technically doesn’t orbit the star, but both bodies orbit their common center of mass. That is, two equal masses will orbit an empty point in space rather than one orbiting the other. It’s just that when the mass difference becomes large enough, that “common center of mass” is located *inside* one of the two objects, which is precisely what happens in the solar system. Because stars are always more massive than planets (almost by definition), a star-planet system can’t have the center of mass inside the planet.
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