Firstly relax everyone, no need to tear chunks out of each other. The anger in this thread is interesting... anyway...
I can understand the frustration for OP, particularly if they have limited time and patience for the situation.
On the one hand, this is a professional environment, and the behaviour could reasonably be seen as inappropriate. Speaking in a baby voice is not something one would expect from colleagues in a professional setting.
At the same time, it sounds as though she may be struggling for any number of reasons, and I suspect this behaviour is a way of managing her distress. People often develop ways of behaving that help them get their needs met, and it is likely that this approach has worked for her in the past. We all learn ways of getting people to back off or provide help when we feel overwhelmed, and I don't think this is unique to her.
What may have been helpful to her in the past, particularly in getting people to back off or avoiding feedback, may not be appropriate or effective in a professional environment. As a child, for example, she may have developed "don't hurt me" behaviours, such as acting more childlike to avoid criticism.
To me, this speaks more to a lack of coping strategies and resilience than to her deliberately trying to be difficult or irritating. I would suggest that compassion needs to be central to how this situation is approached. Most adults would find behaving in these ways embarrassing, which suggests that something else may be going on beneath the surface.
Some people find receiving any feedback that challenges their self-perception extremely painful, particularly when they do not have a strong sense of self. Even relatively minor criticism can leave them questioning whether they really know who they are, or wondering what other people in the workplace truly think or feel about them.
Over the years, I have known many people who, when faced with even the slightest criticism, withdraw and become distant, act childlike, become aggressive, or overcompensate in an attempt to make up for their perceived failures.
The first thought that came to mind when I read this thread was: has anyone actually addressed the behaviour gently and asked her if she is okay?
I might say something along the lines of:
"Sarah, I've noticed you've been carrying a cuddly toy around quite a bit this week, and I'm wondering if everything is okay, as this isn't something that's usual for you."
TLDR: People behave in ways that get what they need or avoid pain. Keep compassionate, as you would hope others would do to you if you were struggling. That includes posters here.