Perfect example, Donald Trump. Somebody who will stop as nothing to dispel any presented threat to their image, name, or brand–usually aggressively. Even if their positive feedback vastly outweighs the negative, they will attack the negative one with all their might until it dissipates.
I know this is a textbook trait of narcissism, but I know a specific word is out there for this. Having 'Thin skin' is the best I can muster informally.
Thank you
Best Answer
Without more context, I cannot pin down which word or phrase it is you are thinking of, for there are endless ways to describe those individuals who “let no slight go unanswered” that risks sullying their self-assessed awesomeness — be it real or imagined, trivial or extreme.
Countless multiword possibilties ranging from biting phrases to hackneyed refrains can easily be found for this. In the arena of single words, there can be nouns for those people, or adjectives describing them or their activities, or verbs characterizing their habitual reaction. Often there will be a cluster of a noun and verb and adjective, where one or two of those derives from the third.
Here are a few nouns and adjectives that come quickly to mind; I give only one form but often there are closely related derived terms for the other parts of speech. For the sake a uniformity of treatment and convenience, all citations are from Wiktionary.