Looking at community law, it looks like they only have the right to ask you to leave AFTER they have trespassed you, before that you have the legal right to be in the store.
Since this lady in fact had not been trespassed previously and they (probably) didn’t serve her a notice during the incident, she would seem to be well within her rights to say prove that it is me, and not leave/call the police to sort it out.
Whilst supermarkets are private property, they are not the same type of private property as your house. They need grounds to refuse entry/ask you too leave, there are all kinds of discrimination implications if you remove the need to have grounds to refuse service.