I bought a new BMW X5 from BMW Park Lane in October last year (annoyed they've just announced a new model, but whatever). They would only allow £6k per car to go on the card, and rest had to be cash. I didn't question if this was just Amex or any card - but they just absolutely wouldn't budge on taking any more.

I think it will depend entirely on dealer and how badly they want the deal.

Side note - what massively helped me was specing up my car, and then going to my local dealer to find out his best price. I then emailed all of the above to all the other BMW dealers to see who came in with the best price. They all actually appreciated the opportunity to compete, and in the end, I got a fantastic deal! You can mention your desire to pay by card, and let them fight it out for your business. Good luck.