This is the only feasible design I could think of when I read the question. Glad to see there’s already a mock up.
I don’t see how it isn’t possible. Obviously there has to be some sort of field that restricts how the blade extends, so why not a field that gives an axe head.
If I recall, and this is maybe just EU stuff that might not be canon anymore, lightsabers formed a sort of loop, which is why they didn't just keep going and going. I am sure there's got to be some kind of way of doing it like an axe, given they have pike ones where it's a long handle. I guess maybe the issue would just be what you'd be getting out of it to make an axe, since a broad blade with shorter reach doesn't really sound ideal for a weapon that doesn't need as much mass. (It'd at least make it harder to bat away, whereas that was shown to be an issue in Force Awakens. Lightsaber versus weapon with physical mass that doesn't just break when hit, equals great difficulty for the lightsaber.)
The field thing, though, is how Gundam does beam sabers, where a field contains the particles and on striking objects, the particles would burn away at what you hit, as you'd expect.