The British 25 mile TT record was set on a fixed wheel.
And Bradley Wiggins was on fixed when he set the 10 record of 17:58
Don't be led astray, however; both records were set by world class pursuiters at the peaks of their powers, and were on their track bikes because they happened to be the best bike they had available, they wanted a training session on their pursuit bike etc. They would almost certainly have been faster on geared bikes, and Boardman admitted that he was spun out coming down the initial hill from Cumnor.
A geared bike is quicker, except for those very few occasions when you're on a dead flat course with no wind and have chosen exactly the right gear, i.e. a day when nobody changes gear all the way round. Then you save a few watts by not wrapping the chain round the tensioner wheels, and maybe a couple more thanks to the cleaner airflow with no derailleurs. Bear in mind that on every other day, people on geared bikes only have to gain about 1% in bio-mechanical efficiency due to optimised cadence to completely wipe out your better machine efficiency.
So, why do I ride TTs on fixed? Well, it freaks out the straight folks, for a start. There's a certain special camerarderie among the small subset of time triallists who are 'on the cog', and you can go up to them in the car park, stare at their their chainring and suck your teeth, making disparaging comments like "You'll be lucky to even get up to the start on that" (H25/2, for those who know it, is my local course)
If I were consistently missing out on the medals by a few seconds, I'd get a geared bike. As it stands, you'll find me somewhere in the third quarter of the result sheet so I'm only racing against myself, and I like riding fixed. It's a harmless perversion, but not a way to win prizes in road time trials.