No. This is completely wrong. Stop misleading people.
The warranty remains valid even if the car is serviced outside the franchised dealer network, provided the work is done in accordance with the manufacturer's specification and genuine parts are used. It's misinformation like this that keeps the dealer servicing costs so high, because there's no effective competition if customers believe they have to use an official Fiat dealer for their servicing. Warranty work is of course different, and you will probably lose any chance of a goodwill out-of-warranty contribution from Fiat, but how often does that happen?!!
Personally, I'll carry on supporting a very good independent garage that actually does a thorough job, while charging me a solid wedge less than Fiat.
Postscript: I see that you are in Holland. The rules will be the same there - it's not a UK thing - because the competition laws involved derive from EU law (the EU Block Exemption rules). So that's another thing that the Great British Motoring Public have cause to be grateful to the EU for.
*ducks for cover*