The former UN ambassador Nikki Haley continued her dogged and faithful defence of Donald Trump on Tuesday, maintaining that the president is a “truthful” person.

“In every instance that I dealt with him, he was truthful, he listened and he was great to work with,” Haley told NBC’s Today. “I never had any concerns on whether he could handle the job ever.”

In fact, Trump’s tendency to be economical with the truth is a much-studied phenomenon. By mid-October, the Washington Post’s Fact Checker column said Trump had made “13,435 false or misleading claims over 993 days” in power.

Tuesday saw the publication of With All Due Respect, a carefully crafted memoir that many people see as an attempt to set up the former South Carolina governor for a return to national political life, even a White House run in 2024. Haley, who left the Trump administration last year, has denied rumours she could replace Mike Pence as Trump’s vice-presidential pick next year.

On Monday night, Rex Tillerson, Trump’s first secretary of state, hit back at Haley’s claim in the book that he worked to undermine aspects of Trump’s agenda in an effort to “save the country”.

“During my service to our country as the secretary of state, at no time did I, nor to my direct knowledge did anyone else serving along with me, take any actions to undermine the president,” Tillerson said in a statement.

“Once the president made a decision, we at the state department undertook our best efforts to implement that decision.”

Tillerson was responding to Haley’s claim that he and the former White House chief of staff John Kelly invited her to join their attempts to subvert the president on several contentious issues, including pulling out of the Iran nuclear deal and the Paris climate accord, and moving the US embassy in Israel to Jerusalem.

In his statement, Tillerson said Haley “was rarely a participant in my many meetings and is not in a position to know what I may or may not have said to the president”.

In her book, Haley claims the two advisers “confided in me that when they resisted the president, they weren’t being insubordinate, they were trying to save the country”.

She also reveals the depth of animosity between advisers surrounding Trump. Tillerson, she writes, “was dismissive of my opinions, and he didn’t make any secret about the fact that he believed his views carried more weight”.

Haley reiterated her accusations on Tuesday, telling NBC what Tillerson and Kelly did was “dangerous” and describing Tillerson as “arrogant and condescending”.

“To undermine a president because you think you know better than him is wrong,” Haley said.

“If you disagree with getting out of the Iran nuclear arms deal, if you disagree with moving our embassy or getting out of the Paris climate agreement, go tell the president. If you still don’t like it, quit.”

In his own response to Haley’s claims, Kelly said that if providing the president “with the best and most open, legal and ethical staffing advice from across the [government] so he could make an informed decision is ‘working against Trump’, then guilty as charged.”

Haley also hit out at the impeachment inquiry, into Trump’s conduct regarding Ukraine, which will bring public hearings on Wednesday.

“Impeachment is serious,” she said, after saying Trump may not have acted appropriately but had not merited impeachment and possible removal.

“The other side of this is we are less than a year away from the election. Instead, let the people decide. Let them hear the testimony, that’s fine, but let them decide.”