Megascale nuclear force
There is no method for building this within the current standard model of physics, but we can step outside of known physics and use cosmological observations to theorize.
If you look at a physical map of the galaxies and galaxy clusters and super-clusters, you eventually get to the point where most of the matter exists in thin walls around huge voids, like soap bubbles. This suggests that there's something in those voids that repels normal matter, but we can't see it, but that's unprovable speculation.
We think of forces as being like gravity or magnetism, where the closer you get the stronger they are, but they aren't all like that. The strong nuclear force, for instance, pulls things together until you get to a certain distance, then starts pushing things away.
Let's theorize an exotic form of matter that interacts with other instances of itself like the nuclear force, but on human scales. Instead of reversing at .8x10^-15 meters, it reverses at something like 10^6 or 10^8 meters. If matter affected by this force gets pushed together hard enough, it will clump. Maybe the core of the planet has a big clump in the middle, held together by normal matter that it's chemically bonded to. Smaller clumps with normal matter attached would be drawn to this core by gravity at long distances, but then stop where the meganuclear repulsion exceeds the gravitational pull.
So, there you go. Floating rocks.