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It might be hard to imagine but then I guess you haven't tried.
I haven't ridden fixed in the peak district, but people have. Like you say, maybe not all routes are feasible but many are, if you are so inclined. (Slight boast post:) I've ridden most of the hills in Kent and Surrey on fixed, many of them are just as steep and twisting as those in the peaks, maybe not as long in places.
Your comment seems a little out of place in a specific thread about fixed gear road bikes on a forum for fixed gear bikes. I'll happily pay you a visit and we can go for hilly ride...
Living in the peak district it's hard to imagine some of these being ridden safely on the roads and hills around here.
The single brake variety mostly seem to be modified track or track inspired bikes which seems to me to be a different thing to a road bike.
Nice as some of them are.