"If you can't find any leaks, it may be the backflow preventer/nonreturn valve in the pump."
That was my first thought as well. The valves in the pump allow water to leak back to the supply side of the pump, the pressure on the output side of the pump drops, the pressure switch turns the pump on, pressure builds, the pump shuts off, then the cycle repeats.
Lack of a working accumulator tank will cause the cycle to happen faster.
Open up the pump and take a look at the valves. If it isn't an easy fix, replace the pump.