"Does he believe the things he is saying because they are completely untrue. I can’t understand how someone could lie so blatantly and not be bothered and he sounds so believable that people believe him."
This sums my mother up perfectly. She almost certainly has Narcissist Personality Disorder and I've been no contact with her for many years. I've never known my mother to apologise or take accountability for anything, It's invariably someonelse's fault.
She rewrites history to paint herself in the best possible light and always as the 'victim.'
She lies about me, my dad, her second and third husbands and her parents. She has no qualms about any of this and is very convincing in her story telling. Her lies reach my ears via various family members who are critical thinkers. Lies she tells about me no longer bother me, I just roll my eyes and shake my head. But lies she tells about my dad, (far from perfect but not the monster she paints him as); and my late grandparents, (lovely, kind, honest people), get my back up. More than once I've defended my dad and my grandparents by relating the truth often by disclosing unflattering information about my mother's actions which she, ofcourse, conveniently chose to leave out of her revised edition. In years to come, your child will probably be doing the same.
In addition, it's a very dangerous game your ex is playing badmouthing you to your child. My mother did the same thing to me from the age of 7 (she called my dad a "bastard" amongst other things) whereas my dad has never said a bad word to me about my mother, something I've always respected him for.
In my experience, there's not much you can do. If you challenge a narcissist with the truth, they'll deny it, fly into a narcissistic rage and call you every name under the sun. They'll then go on a proactive damage limitation exercise presenting themselves as the 'victim' by telling anyone who'll listen what an awful, cruel person you are. Some will fall for it, others won't.
Where your child is concerned, I'd handle it by looking bemused.
Child "Mum, dad said x, y and z"
You (puzzled expression): "Your dad said x, y and z? Really? Mmmm..."
Then change the subject.