South Dakota Republican Governor Sources close to Donald Trump tell The Post that Kristi Noem “has no shot” of being his running mate, after she revealed in a book coming out soon that she had shot and killed her dog.

Noem, a former Republican candidate and current author of “No Going back: The Truth about What’s Wrong With Politics and How to Move America Forward,” was described as “bewildered” by the team of Trump when they heard of Noem’s account.

Noem was not expected to be Trump’s VP but she had a chance.

After this, it is just impossible.

Noem wrote, according to The Guardian, that Cricket, a wirehaired 14-month old pointer, was put down after she determined that her “aggressive” personality made her “untrainable”.

The 52-year old recalls that “it was not an enjoyable job, but it had to get done.”

The source told The Post that Mitt Romney, the 2012 Republican nominee, is still remembered as having put his dog into a crate on top of a family station wagon during a 1983 journey from Boston to Canada.

This person stated that “Trump doesn’t like dogs necessarily, but I think he knows you can’t pick a puppy-killer as your choice, for obvious reasons.”

A second source told The Post Trump “likes Kristi very much,” but was “disappointed” when he heard the “dog story.”

This source said that “it hasn’t improved her chances but no decision has yet been made about any of the vice-presidential candidates.”

Noem, who is now in her second term of office as governor of South Dakota, recalled that Cricket’s final straw came when she saw the dog jump out of the truck, while visiting her neighbors.

Cricket, unfettered, attacked the chickens of the family, “grab[bing] each chicken, crunching them to death with a single bite, and then dropping it to assault another”, according to the Governor.

Cricket was the “picture of pure joy” when he “whipped around and bit me” as Noem attempted to restrain the canine.

Noem describes in graphic detail the killing of a male goat that was “nasty” and “mean” because it wasn’t castrated and “loved” to chase her children.

She had to repeatedly shoot Cricket to kill him.

Noem wrote that she told the grisly tales to demonstrate her willingness to take on necessary tasks that are “difficult and messy” — but admitted that, “I suppose if I was a better political I wouldn’t have told the story here.”

In the face of animal rights protests, the governor has since reaffirmed her decision.

“I understand why some people were upset by a story from my upcoming No Going Back book, which tells the 20-year-old story of Cricket, a working dog at our ranch. Noem, who wrote on X-Sunday, said that the book contains many stories about my life. It includes good and bad times, challenges, painful choices, and lessons.

South Dakota law stipulates that dogs that attack and kill livestock may be put down. “I decided to do what I did because Cricket was aggressive toward people and bit them,” she said.

Noem’s latest controversy is about the animal-killing story.

The Post, among other publications, reported in September that Noem was involved in a long-term affair with Corey Lewandowski, a Trump advisor who has been around for many years.

The governor of Texas was sued last month for “misleading” and “deceptive” advertising. She had endorsed the Texas dental office that installed her new veneers.

Trump has not yet revealed who he will choose to be his No. Noem was No. 2 on his list of potential picks. She had a strong relationship with Trump and previously stated that she would have accepted the position “in a flash.”

The Post’s inquiries to the Trump campaign and Noem’s office were not immediately answered.