Here's a great way to understand why your mother lies about you: she's projecting her own issues onto you.  She can't admit what SHE does, but if she makes it look like YOU are the one doing these things, then she can get out her self-directed aggression through you.

Right now my mother is going around telling people that I have no friends and I can't hold onto friends because I am always mad at them.

Funny, because that's exactly what she does.  She goes through cycles of hating the holy hell out of her friends (and doing terrible things behind their backs during this time) and cycling back to put them on a pedestal (like they are the BEST THING EVER!!).  This is called "idealization and devaluation" or "splitting".  She throws away friends like TP, and then reels them back in with praise and insanity.  She did this regularly with me, sometimes quickly, and sometimes both would take months (just enough to trick me into believing she'll be good for a long time).  The change between the two is INSANT.

She's now equating those same issues onto me, saying that I throw away my friends like garage and I've never ever been able to hold onto any friends.  Like, ever.

Yet, I had the same friend from 4th grade until I was an adult.  And I had the same BFF's in high school until I became an adult (until I got pregnant and my very close BFF went around telling everyone we knew that my ex wasn't the father of my child...which was a bullshit move--I later did some soul searching and realized she was also a narcissist, which made me not angry with her anymore).  As a kid, all my friends were my friends until they moved away.  Actually, I kept most of my friends throughout my life until that reason: either moving away, going away to college, etc.

Yes, as a teen, I fought a LOT with my BFF's, but we stayed friends until my first pregnancy.  All I grew up with was drama and fighting, so that's all I knew.  I was attracted to drama and created it wherever I went.  But I eventually grew up and realized drama was for losers (aka. my mother) and eliminated anyone from my life who were drama queens (or kings).  It's been a slow journey...with lots of mistakes along the way, but here I am.  Friendless.  Hahaha!

So she's right in the respect that I can't hold onto friends, if you count putting up with 30 years of total drama and bullshit from those I've known since grade school, middle school, and high school and then saying goodbye to them as an adult because I realized our friendships just weren't based upon adult interactions.  I guess that's true then LOL  But I don't throw away friends like garbage.  I may meet someone, realize I don't like them, and then move on from them, but they aren't actual friends.  They are just people I know. 

Whereas my mother will take people she supposedly CARES about and shit all over them until they either walk away or get lured back into her crazy and fucked up worth via the "ideation" stage.  She literally throws them away in a moment's notice, for NO reason (sometimes for a made up reason or just a little itty bitty reason) just so she can get her narcissistic supply from the situation.  Fuck the actual person...who cares about them?  They are just her tools.  Like when  hammer is of no use to a contractor anymore, it goes in the garbage.

I just find it funny how she projects her own bullshit onto me.  And I guarantee you that your mother is doing the same thing when she's running around exaggerating or lying about you.  So instead of getting angry, try to see where her lies are coming from.  Most of the time you'll find they are coming from her own mirror 😉