How old are these children? It’s a storm in a tea cup- one child tells another her pet is the best/better than the other child’s pet is hardly uncommon behaviour in children (even if it is a little unkind). Your DD is going to come across similar situations in the future, because this other little girl is not behaving way off the scale of usual childhood behaviour.
Don’t you think that you need to model how to handle disagreements- even one where feelings are hurt- in a reasonable manner, rather than jump to the nuclear option? I think if your first reaction over your DD being upset is to decide permanent NC- especially given it is a family member of your DP’s- you are significantly over-reacting. She doesn’t need to be this girl’s best friend, but there is a stage in between that and NC, surely? I think you are over-reacting to your upset at seeing your DD’s feelings being hurt- you need to put your feelings to one side and help your daughter learn how to manage this situation well, rather than ramp up the volume unnecessarily.
If you don’t manage this sort of thing properly, your DD won’t develop the social skills to manage disagreements/conflicts etc appropriately- it’s something we all need to learn to navigate. That doesn’t mean ignoring the upset or teaching her to be a doormat. Surely, the best way to deal with this would be to acknowledge that this other girl has said something somewhat unkind about your DD’s pet, which wasn’t a particularly nice thing to do, but reassure her that as long as DD thinks her pet is wonderful that is all that really matters in the end. I’d explain that the best thing to do is ignore the girl’s comments, or simply state that she thinks her pet is amazing and leave it at that- and going forward she doesn’t have to be friends if she doesn’t want to be, but they can manage to be in the same room as each other in the future.
Life is going to be infinitely harder if she learns that the only response to someone upsetting you is to cut all contact with them, rather than either try to resolve the disagreement, or maintain a polite distance if that can’t be achieved. NC should only be a last resort after repeated problems, or after a really significant one. A child saying her pet is the best really is not in the latter category.