https://numericaltank.sjtu.edu.cn/three-body/three-body.htm shows that there are three dimensional solutions.

We have no observed examples in nature of three body equilibrium. But then again, all places we have looked are either influenced by the chaotic orbits around them of the Solar System, our surrounding galaxy, or nearby galaxies in a cluster.

There aren't a lot of orbiting three bodies without external gravitational influences disturbing them.