I recently got my car serviced and I need a new battery. My old battery is still under warranty. Should I be able to claim a refund on the cost of the new battery?

 Your old battery would need to be tested at the supplier you bought it from before you buy another one. The outcome and terms of your warranty will determine whether a claim is possible.
If your car battery stops working and is still under warranty, contact your battery supplier first. They’ll test the battery for you and if it’s faulty they’ll replace it for you according to your warranty.