You have a problem with that wiring or something downstream of that outlet.
This is where I would look. A GFCI can function entirely without a "ground". The lay person usually thinks of "ground" being the equipment grounding conductor (green wire). There are actually two "grounds" in modern wiring, although one of them is allowed to be a current carrying conductor and the other is not supposed to be. Your grounded neutral conductor could be touching an equipment grounding conductor somewhere downstream that would trip out a GFCI or AFCI. Somewhere, there is a fault that is allowing power to flow from the line side of your GFCI to somewhere other than the neutral of that same GFCI, and it is causing it to trip.
Not implying you are the lay person in this thread pcgeek11, as I know for a fact it is the contrary.
Basically, OP, the GFCI is doing what it is designed to do, and you either need to figure out what is tripping it, or hire someone that can fix it. Pests chewing on wires is a very unlikely possibility, but they will definitely not start a fire as the GFCI is cutting the circuit out like it is designed to do.