Is there a maximum airfield elevation in which a Cessna 172P can operate?
You should look in the POH for your specific aircraft (because it may have had modifications to it which would make it different from a "standard" aircraft) and see what it says.

The 172P POH that I found online contains no maximum airport elevation for takeoff, and the Type Certificate Data Sheet (TCDS) doesn't either. 

Therefore, you can only go by the performance charts in the POH.  According to the POH that I found above, assuming that you have enough runway available and can out climb any obstacles (big assumption, you need to make sure of this), you can takeoff at a pressure altitude of up to 8,000 ft. other than in cases where the airplane is very heavy and it is hot, in which case it will be restricted further.

This doesn't mean that the aircraft is not capable of taking off at a higher pressure/density altitude however, and if you have the need to do so then I would recommend contacting the manufacturer directly to see if they have any engineering data available for your situation (often they do but don't publish it; there may or may not be a fee).