How to deal with a team member who tries emotional blackmail on us?

We have a team member who has to create some input for a project we are working on. This team member is terribly sensitive to even the slightest criticism. They started a discussion how they feel slighted. I have a leading role in the team, although I don't have a high managerial role in my organisation's hierarchy. The team member is not my subordinate, they have been loaned to the project in a supporting role. When the team member complained (of a situation in which I was involved), I tried to be fair, accepting and understanding. I apologized for not having considered the impact on their feelings before I took action back then, and reversed my earlier decision. By the way, the decision I had taken was not so unusual, I have made similar ones in other projects without team members complaining, and a senior member of the team confirmed that while the decision fell in somewhat of a grey area, it was not obviously incorrect or insensitive, and the consequences are so minor that it shouldn't have caused drama at all.

The team member accepted my apology. And then they launched the next complaint, about a different situation. This time, they had submitted a document we needed, and somebody from us asked for clarification. It was a terse question, which can certainly be interpreted as a challenge, on the lines of "Here, why do you write that so-and-so?". Back then, the team member had given some answer and the communication about this issue had stalled, mostly because of lack of attention to this detail. Now they informed me that they have removed their document from the shared workspace storage, and that they are considering leaving the project, because they are getting the message that their collaboration is not appreciated and they are unwanted. They have removed a document before in a similar situation, but the "I feel pressured to leave" theme is new. I certainly would like everybody involved to feel good, and I am aware that more politeness can certainly lubricate the communication. But my view is that this person is badly overreacting. Sure, they don't want to be criticized, nobody does. But saying what amounts to "I will take my dolls and go away if you bad people make me cry", especially when the threshold of them feeling threatened, unwanted, etc. is so much lower than that of most other people I have worked with, is not constructive adult behavior. And sometimes negative feedback is needed, because their work actually needs to be improved now and then. The work of the others (including mine) needs it too, and the frequency and presentation modus of the negative feedback in the project seems to work for everybody else without causing resentment.

In many situations, I find that a frank personal talk about any issues is a good first step to take, but what can I do when even the whiff of criticism makes the person fall into a highly emotional defensive mode, demanding apologies and reassurance that we are not trying to get rid of them? We don't actually have to work with this person. If they decide to leave, we could get a replacement to supply the same input, or quickly wrap up the project. Still, I find the social situation very intriguing. Is there a way to improve the situation so we can continue to collaborate in a professional manner? Beyond walking on eggshells and reassuring them on every step that we are not mean to them? Is there a way out beyond them parting ways with the team?

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