Hi Anne Marie,
You’re in a great group of people here to ask this and I’m sure lots of answers will come your way soon. Being breastfed till 3.5yrs in my eyes is just amazing! You are a trooper to give so much of yourself to this beautiful child.
I’m no expert by any means, so I can only offer what seems obvious to an untrained eye. Either your daughter has a vitamin or mineral deficiency which needs to be ruled out by your pediatrician or its behavioural. Some deficiencis can cause unnatural cravings to eat unedible thing.
If it’s behavioral, then I’d read up on the HiH article about stuttering. Totally different problem, but I think the solution lies in ignoring the behavior to not add our own anxieties to the situation. Kids pick up on our feelings like radar patrol. Some listening time with your partner about how you feel about getting sick from the cold or perhaps how you felt about yourself as a child with any lingering baby like habits.
It’s also quite likely it has something to do with the weaning from breastfeeding. Maybe a habit has replaced that comfort for her. In that case, you could read articles about ending habits like pacifier. You can tell her we won’t be chewing on anything today as a gentle limit and then do stay listening when she shows upset about it. Though I’m really not sure if it’s the right route as the example of stuttering would indicate to not draw too attention to the issue. I think it’d be wise to ask one of the HiH consultants on this to be sure or join a weekly call to ask.
Wishing you and your little girl the best,