Specifying thickness, material and quantity could help us with suggestions.
It really is a different story if you need 200 or 200 000.
Anyway, when I needed some thin, but large plastic washers with small hole, I used a lathe to put a hundred or so roughly cut pieces on a mandrel and turned them all at the same time. They had to have a small hole in the middle, though (but in my case that was needed anyway). If you can live with a 5 mm hole, the process should be possible for you as well.
If not, or you don't have access to a lathe, and need fairly small quantities look into having it laser or waterjet cut.
Laser is fairly cheap, but some shops may refuse to cut copper alloys. That was the case when I had to have some things cut from bronze sheet (something about reflections damaging the laser).
I ended up waterjet cutting. There is basically no restriction on material there, but it is a bit more expensive.
If you need large quantities, punching will be much cheaper, after the initial setup fee.