First you need to know if it is a PVD or a "living finish". But anyway you look at it your more than likely out of luck and the finish is ruined.

Replace it. It's ruined.

I think you will need to replace it. I got rust on the drain round piece of my sink in my bathroom and it spread terribly. I think rust must be something that is living and spreads and is difficult to remove. I have to now remove parts of the sink and most likely replace the sink. The faucet was newly installed in 2006 and I told the plumber to replace all sinks but he felt the sink was in good shape n my Master Bathroom and said that I would have to replace both sinks to have the sinks match and he felt the rust would not spread as it was tiny then and now has permeated and has damaged the sink. I should have made him remove it then as it is costly to keep paying for plumbers to visit.