no they are not flashing but they never did since I got the car .but central locking worked for the first three years and in the last six months it doesn't.but lights have never flashed.im from Clydebank near Glasgow
Doesn't look like the central locking has a single fuse, problem doesn't sound like this though.
It sounds like the key has lost its communication with the car (due to flat fob battery). This can be re-done to the car via MES (Multiecuscan) IF you have the security key code for the car. Or a Fiat dealer can do this (2 minute job BUT take ALL your keys with you to make sure all are done). Not sure how much Fiat would charge for this. MES or Fiat can also turn on the indicators flashing when you lock/unlock the car also.