Donald Trump has called former congresswoman Liz Cheney a radical war hawk and said she should face being under fire with rifles “shooting at her” – prompting the anti-Trump Republican to warn the public against dictatorship and a presidential candidate who “wants to be a tyrant”.

Cheney recently endorsed Kamala Harris and has campaigned with her, trying to persuade Republicans who don’t want Trump to win another term in the White House in this election to vote for the Democratic ticket of the US vice-president and her running mate, Tim Walz.

Harris on Friday said Trump’s violent rhetoric about Cheney “must be disqualifying” ahead of the 5 November presidential election. Meanwhile, the Arizona attorney general, Kris Mayes, on Friday said her office had opened a “death threat investigation” surrounding Trump’s comments about Cheney.

While still in office, Cheney co-chaired the bipartisan special House committee that investigated Trump’s conduct on January 6, when extremist supporters of his stormed the US Capitol to try, in vain, to stop the certification of Joe Biden’s electoral victory over him. She lost her Wyoming seat in 2022 as Trump supporters turned against her. Trump has called for her to be jailed for investigating him.

Trump was in Arizona on Thursday evening, doing a sit-down talk with the conservative broadcaster Tucker Carlson, and he brought up Cheney in the context of her father, Dick Cheney, the former vice-president who was a hawkish architect of the US invasion of Iraq in 2003 during the George W Bush administration. Trump has been critical of that war and has also criticized the current US president, Joe Biden, for becoming involved in Ukraine’s struggling defense against the invasion by Russia.

“She’s a radical war hawk,” he said of Liz Cheney. Then said: “Let’s put her with a rifle standing there with nine barrels shooting at her. Let’s see how she feels about it. You know, when the guns are trained on her face.”

Early on Friday, Cheney issued a stinging response on X.

“This is how dictators destroy free nations. They threaten those who speak against them with death. We cannot entrust our country and our freedom to a petty, vindictive, cruel, unstable man who wants to be a tyrant. #Womenwillnotbesilenced #VoteKamala,” she posted.

Trump made his comments on Thursday about Cheney after complaining that Democratic criticism of his campaign had fueled what according to authorities were two failed assassination attempts against him.

Mayes told Arizona’s 12News it was not yet clear whether Trump’s comment amounted to protected free speech or a criminal threat.

“That’s the question, whether it did cross the line. It’s deeply troubling,” Mayes said, according to Reuters. “It is the kind of thing that riles people up and that makes our situation in Arizona and other states more dangerous.”

Cheney has said she has never voted for a Democrat before but will do so “proudly” for Harris in this election to ensure Trump never holds a position of public trust again. Her father will join her in casting his ballot for Harris.