I have a GFCI in my kitchen that seems to keep making a clicking
noise. The GFCI is connected to one other outlet along the
countertop. The funny thing is that it keeps doing this every 15
minutes or so even after it has tripped off (without me reseting it).
This is the same noise as if it is tripping from the on to the off
state. This is all happening while nothing is plugged into the
outlets.

I have tested that the GFCI is working by plugging a light in and the
light works when the GFCI is set, but turns off when I press the test
button.

Is it a bad GFCI outlet?

Is this some way of indicating to a person that it has tripped by
making the noise?

Is there something wrong with the wiring that I need to take a look
at?

Any advice or other thoughts are appreciated.

-DK