Abusive sibling. Any advice?

Tell her it’s your home and if she tells another lie she won’t be welcome in your house anymore.

Try to record all interactions with this sister, particularly between her and your mum, if possible. Your sister is emotionally abusing your mother. Do some research into narcisssistic abuse. It's where someone tries to paint a picture different from reality in order to make themselves look good and to destroy you as a person, in effect.
Most likely, professionals may not believe you as people who carry out this abuse will deny everything and claim that they are the victim of your abuse. They can be very convincing. You may even be accused of abusing your mother so the key is to know what you are dealing with. My sympathies to you as my own sister and her husband abused my mother in this way and convinced professionals and mum's family that i was the abuser. It is truly evil to treat anyone, let alone the elderly in this way.

Try to talk via text. That way you have proof if she is harassing you. Tell her no more talking by phone.
Id still block her from your house if she can't be civil. You and your mom don't need that stress.