Will Apple replace my broken Lightning cable for free?
I have had the same situation happen to me with 2 phones actually.  Apple did replace them under warranty.

If you have access to an Apple Store:


Either walk in or make a Genius Bar appointment
Be sure to have your iPhone with you
Apple will need your IMEI number from your device


To get the IMEI number go to Settings > General > About (The Apple rep can assist you)



They will then provide you with a new cable after verifying your eligibility.

If you do not have an Apple Store that you can get to, call AppleCare for instructions.