Well pump exceeds normal pressure
This would appear to be most likely a faulty pressure switch.

Start there.

The job of the pressure switch is to turn on when the water pressure is low, and turn off when it is high enough - this is most commonly a 20 PSI range (ie, 20-40, 30-50 or 40-60 PSI.)

The pump is evidently working. 

The pressure gauge is evidently working unless you see some other behavior that suggests otherwise, such as a non-zero pressure when you drain the system.

A tank problem would not cause uncontrolled pressure rise. The most common tank problem (failed bladder/waterlogging) would cause abrupt pressure rise and the pressure switch would cut off very quickly.

Unless you have gremlins that would have altered the adjustment of your pressure switch, all reported symptoms point to switch failure. Start with replacing it, should be in stock at most any hardware store.