Our 2014 Countryman S automatic seems to not like our brand new Garmin GPS. Every time I plug the Garmin into the lighter socket, it makes a repetitive "clicking" sound about once every 10 seconds. Almost like something is tripping, like a fuse circuit, however, the GPS does not seem to be affected (it stays on and operational). I'm worried that keeping the GPS plugged into the lighter socket may damage something electrical... or at minimum blow the fuse associated with the socket, or blow the fuse that's inside the GPS cord. (It's a new Garmin Nuvi 3597).

Any idea what is happening here?

By the way, this GPS works perfectly in our 2006 Honda Pilot and 2006 MCSa with no clicking sounds at the socket/plug base.

 

I have not had any issues with my Garmin and the CM. I assume you are using the Garmin cord supplied with the unit since you cannot simply hard wire it directly to the fuse box. I believe the 2014 uses a 20 AMP fuse for lighter socket (it assumes 10 for the front outlet and 10 for the rear socket).
You could try isolating whether the issue is with the socket or the garmin.
1) Since one fuse supports both sockets/outlets, first check to see if another accessory works in the CM. Try the front socket and the rear.
2) I would try the rear outlet to see what happens with the Garmin. If the Garmin charges with no issues in the rear socket, I would check if anything is in the socket (look inside).
3) You could try to simply removing the fuse and reinstall it in the Garmin outlet base. As this hard to watch video shows.
http://www.youtube.com/watch?v=uuT4JURXKM0

Last, I am assuming you did not use a fusetap to connect any additional accesories into that same fuse. If you did tap the fuse, you need to make sure the ground is attached its own metal screw somewhere to avoid a ground loop.