You need a scalpel or a Stanly knife. Make two closely spaced parallel cuts on a track and then remove the copper in between the cuts with the blade.
Or you can use the tip of a twist drill turned by hand in the hole. A tool that does the same job, but better because the angle on the tip is more suited for the job, is called a "spot face cutter". spot-face-cutter
If I want to break the connection between, say, pin 1 and pin 3, I use a 3/32 cobalt drill to remove the pad at pin 2 since this is easier than the razor blade method. Also, I spin the drill by hand. It would grab the board and make a 3/32 hole if i used a drill motor.
Note to everyone: The PC board is fiberglass is VERY abrasive. It will eat ordinary high speed steel drills for breakfast.
Even titanium nitride coated drills don't last long. A cobalt (not to be confused with the BRAND "Kobalt") drill lasts a bit longer, but the best is a solid carbide drill. These cost more, but the fiberglass doesn't dull them.
Carbide drills are very hard and very brittle and there is a 99% chance that you will break one in a hand drill or drill press.
But, for hand held deburring of a hole or cutting a PC board trace, they are ideal. Don't drop it though... it will shatter like glass.