Not on fixed, but the LEJOG record (bicycle) is held by Gethin Butler with a stonking 44hours 4mins and 20seconds, but that was supported and with gears.
Andy Wilkinson holds the overall record using a faired recumbent, also supported, with a time of 41h 4m and 22s.
There's no specific fixed wheel record listed, or distinction between supported and unsupported, so he may have a claim to it but only because no-one has claimed it before.
It's also worth noting that the standard time limit for a LEJOG under Audax UK rules is 4 days 20 hours and I'd be really surprised if someone hadn't completed that on fixed with more than a few hours to spare.
Fixed isn't that much of a hindrance if at all, it depends more on the weather than the terrain. My fixed/geared average speeds are almost identical on similar rides over 200km, including hilly jaunts through Wales.
I'd expect a strong rider could do a sub 48-hour fixed LEJOG.
In general, the strong riders go for the all out unrestricted, supported, records rather than the specialist ones (fixed, unsupported, mustachioed, etc).
I'd be looking at taking the full 116 hours 40 minutes that Audax UK would give. (Actually, if I did LEJOG I'd like to take my time and see the scenary rather than just slog through it for 5 days.)