I have the same problem with one of my garages. I remember reading about a track-style unit, where a very thin track goes over the open garage door. The motor unit follows the track, instead of remaining stationary. Therefore, there is no chain or other stuff along the track. ½” clearance is all that is needed.
Now the bad news. I have been unable to locate it, either locally or through the Internet. The links Ed sent offered no help (sorry, Ed). All they could suggest was for me to replace the garage door with a shorter unit, leaving more clearance. However, I would loose about 3” of headroom when the garage door is open. I did not want to do that since my garage is already lower than normal.
Now, some good news, at least for me. Maybe you can do this as well.
Since my garage is real deep, there was room to mount a standard garage door opener behind the farthest point the garage door is when it is opened. In other words, it is mounted on the ceiling as normal, except it is behind the opened door. I simply fabricated a 7’ pole, made of conduit tubing, to connect the door to the opener. Works fine. In my case, my doors are on a double roller track since they are low-clearance, so the opener mounts on top of the top panel instead of the middle of the top panel. Therefore, the pole I made stays high on the ceiling and does not get in the way, garage opened or closed.