I dunno about a forklift, but a tractor I got unstuck...
You'd need to check your lift/push ratings, but a variation on this might work:
A few decades back I (or was it the other Ranger?)(it was a long time ago!!) was driving a tractor (with a fairly crude, lightweight front bucket) in some back country when both front wheels dropped into a rabbit warren. By 'dropped' I mean, front wheels past the axles, engine/gearbox stopped it falling further. Must have been a series of large cavities in dry soil.
There was no way a 1-tonne 4WD or a winch was going to pull it out.
All I did was lower the bucket to the maximum extent possible - raising the wheels out of the cavity - and reverse with the bucket as a slide. Out in moments. The other, inexperienced Ranger, most impressed.
My thought is, the actual forks, while designed for lifting 'up' at front, might be used for pushing down on some hard surface, big old planks or something - and if you can't lift both wheels (you MUST check the rating, see if this is a Bad Idea for a forklift!!) maybe you could tilt/lift one side, till something super-solid can go under the driving wheels (and the steering wheels as well, if possible) - it would then be a business of walking it out over one plank or expanded mesh length at a time.
Emphasising again: it worked for a tractor with a slightly clumsy, after-market bucket/front-end-loader, (and it's the back end of tractors that's heavy, not the front, even with a bucket) - but it might be a method of moving the forklift to parking on a reinforced concrete slab.
Note also: I have seen parked vehicles on thin, hot bitumen leave both dents and crime-scene casts of where they were parked.