I went to plug in my outside XMas lights yesterday for the first time - my exterior outlets (with GCFI test/reset button) is tripped and does not work. The test button does not push in, the reset button does, but does not pop the outlet back on. I have gone thru my whole house with a fine toothed comb pushing every RESET button I have.. They are all working. This house was built 10 years ago, so it is not like there is old wiring. I have no idea how long the outlet has not been working - I only use it at XMas. Worked fine last year. Not only that but when I check my circuit panel there is not a breaker for the outlet ???? Please Help !! Thanks!!
GFCI outlets do go bad especially outside. There should be a breaker some place in the house if not in the main panel there could be a sub panel. I would be amazed if you throw the main breaker if there is still voltage. Non contact voltage testers may indicate a live circuit When the breaker is turned off because of induced voltage from another cable run in close proximity to the circuit you are testing. With the main turned off you won't get the phantom indication on a non contact tester.
What you are describing is exactly what you would expect if the GFCI is not receiving power.
If it were just tripping because of an actual ground fault the reset would work with a click then it would pop again.
I would recommend you go measure the voltage on that line at the breaker box just to make sure the breaker itself is good.
Then if it is, pull the GFCI out and measure the voltage on the wires there.
From that you can determine whether the problem is at the GFCI itself. Like @Ed says, they do sometimes just go bad.
If there is voltage at the wire, then you can replace it and see if that's all it was.
If all that doesn't work, time to call your best electrician friend.