While not exactly the sub-moon you're looking for there are some interesting objects that come close.

Co-orbital moons, like Saturn's Janus and Epimetheus, are moons that are in stable orbit-swapping configurations. https://en.wikipedia.org/wiki/Co-orbital_configuration#Co-or...

Then you've got asteroids that are contact binaries that also have moons, like 216 Kleopatra. https://en.wikipedia.org/wiki/216_Kleopatra

And finally, you have 624 Hektor, which is a Trojan of Jupiter (stable solar orbit near Jupiter's L4 Lagrange point), that is a a likely contact binary that has a moon. So it's orbiting the Sun, while "orbiting" Jupiter's L4 point, while all 3 components (the two halves of the contact binary, and their moon) orbit the center of mass of their own little 'Hector system'. https://en.wikipedia.org/wiki/624_Hektor

It's theoretically possible, but not likely. The Earth-Moon-Sun system is fairly stable because the Earth and Moon are relatively far away from the Sun and everything else. When we put a satellite in orbit around the moon, the gravity of the Earth is much greater when the satellite is on the Earth's side, and much weaker when it's on the other side of the Moon. Those forces destabilize the orbit rather quickly and corrections have to be made using fuel.

If the Moon were much further away, the Moon could have a moon of its own. But the further away it gets from the Earth, the more likely it is to be influenced by other objects like other planets or the Sun.

Apparently no [0]

[0] https://earthsky.org/space/can-moons-have-their-own-submoons

Not that we know of, but there are asteroids in the belt that have their own satellites / are binary systems.