Well, after the incident with the mud (see photos a few posts above in this thread) I took the knobblies back off. I fitted some 25mm Continental Gatorskins. (I had put 28mm Gatorskins on my geared gravel bike.)
The Gatorskins are lovely tyres, handle well, and have excellent puncture resistance. They have virtually no tread pattern, but that's OK as I am a bit sceptical about tread on narrow bike tyres anyway. All it seems to do is pick up mud, without providing much grip, and it makes road riding hard work.
So, today I took the Pearson out to Sherwood Pines. This is an area in Sherwood Forest, Nottinghamshire (England) where there are footpaths, bridlepaths and official mountain bike routes. I used to ride unicycle there regularly. The paths offer a mix of packed mud, soft mud, sand, loose gravel, tree roots, occasional outcrops of rock, short steep climbs and descents, and undergrowth.
With me on the 700c x 25mm fixed road bike, my wife on her 700c x 23mm geared road bike, and my son on my 700c x 28mm geared gravel bike, we rode a wide selection of tracks and trails, sometimes riding gingerly over obstacles, sometimes blasting along, and on one heart-stopping occasion, sliding down a steep muddy hill with the back wheel locked and carefully feathering the front brake.
Funny thing was, on my skinny fixed gear road bike, I think I was having more fun than all the people on dual suspension multi-geared mountain bikes who had little to do except point the bike where they wanted to go and let it find its own way over the bumps. I had a great time.
The scenery is varied, but all beautiful, with areas of beech wood, areas of plantation pine, oaks, and mixed woodland, and a wide variety of birdlife. I saw several male bullfinches (splendid little birds with red fronts and black caps), a jay, wren, and many buzzards, magpies, crows and wood pigeons. In the past, I have often seen wild deer here, but not today.
So, in answer to the original post, yes, a fixed can be ridden on challenging off road conditions, and is terrific fun. My bike is not optimised for this (limited clearance, low bottom bracket, rim brakes) but the only problems I had were two pedal strikes. However, my son fell full length from my proper gravel bike, at low speed, after getting is feet stuck in the toe straps. How we laughed.