South Dakota's political leader tells the harrowing story in her upcoming memoir, "No Going Back: The Truth on What’s Wrong with Politics and How We Move America Forward" ... discussing farm life and detailing exactly why she says her young wirehaired pointer had to die.

The Guardian reviewed the book and rehashed this sad chapter. KN owned a 14-month-old pup named Cricket with an untrainable and aggressive personality, and didn't turn out to be the best hunter either. After an attempted hunt, she says she started driving back with Cricket ... only to make a stop at a local family's home, with the dog attacking their chickens.

The dog apparently killed some of the chickens and when Kristi tried to get it under control ... the canine nipped at her ... and that's when she realized she had to put it down.

Noem says she grabbed her gun and led Cricket to a gravel pit ... where she shot the dog to death. She adds she took no pleasure in the act, but claims it was necessary.

The governor -- who many believe might be Donald Trump's VP pick -- adds she's had to kill other animals before too ... like an aggressive goat that knocked her children down because she says it wasn't castrated.

She shot it in the same gravel pit as Cricket ... but says it took two bullets to kill the goat because she only injured it the first time.

As you can imagine, the outrage on X is fierce ... many are blasting Noem for killing defenseless animals, while others are sharing pics of their dogs in solidarity with Cricket.

Kristi's not bothered though, responding to one outlet's article by saying animals die all the time on farms ... and, encouraging supporters who want to read more wild stories to buy her new book. Unclear if this tale is gonna affect Trump's decision in possibly selecting her.