There are discounts available from Fiat dealers, without a doubt. I saved 6.5% on mine through being a member of the CSMA. More than one dealer I approached when shopping around seemed to think that in fact they could sell the car for even less, if it meant they got my business. And that was 2 months ago when the official line was that the principal of transparent pricing meant that the price you see is the price you pay.
Nonsense. Don't expect much, as the cars genuinely are very good value, but discounts are available to those who ask. Make it clear that you will buy a car immediately if you hear the right figure being mentioned, and be prepared to put down that deposit right away.
Also, another way to save money if you don't want to pay in full up front, is to haggle the finance down. I've heard of deals on Fiat HP of around 3%, They'll tell you that they'll charge more, but this again is nonsense. If you don't ask, you don't get. At first I was told that I would not get low rate finance because I had a discount lined up. The next dealer I went to said they would commit to both the discount and the low rate finance. I went back to the original dealer, told them this and they said they would do the same, on the spot.