Mr Musk, 52, and the New York congresswoman, 33, have been trading barbs this week after the Tesla CEO criticised the congresswoman’s view on immigration.
In a video posted on X, formerly known as Twitter, Ms Ocasio-Cortez claimed more migrants entered the US through Ellis Island than were arriving illegally across the Rio Grande.
Responding to the video, billionaire Elon Musk said: “She’s just not that smart.”
The congresswoman clapped back: “I wasn’t born rich and became the youngest woman in American history to be elected to Congress. Then I investigated Cohen, authored the largest FEMA funeral assistance program in history and led creation of a US Climate Corps to create tens of thousands of new jobs.”
“Stay mad,” she told Mr Musk.
The Tesla CEO replied: “I take it back – you’re obviously a genius.”
He added: “First, please allow me to congratulate you on these epic achievements.
“However, have you considered, rather than ‘funeral assistance’, that FEMA focus its funding on preventing funerals in the first place?”
The row later escalated after Mr Musk paid a visit to the US-Mexico border and attempted to live stream the experience on X.
AOC accused the billionaire of taking a “joyride” to the border, after he traveled to the Texan city of Eagle Pass and claimed some “pretty extreme people” were arriving illegally.
Mr Musk said he was concerned about stories of violent criminals finding their way into the country among those fleeing poverty and conflict.
“We are in this absurd situation where the vast majority of people are not in fact asylum seekers - you can actually read on Google in any language you want what are the magic words you need to say,” claimed Mr Musk.
“And then you are automatically in the asylum, sort of, queue. Which takes several years before you actually see anyone. And then even if you do see one, even if you are denied, you are not deported.”
He added that he is “very much in favor of expanded legal immigration.”
“I think anyone who is hard-working and honest and will be a net addition to the economy - I think we should let them in,” he said.
“But I mean what we are seeing are, in some cases, some pretty extreme individuals coming through. Obviously not suggesting everyone is like this.
“They have murdered someone and they are so proud of having murdered someone they tattoo one tear on their face for every person they have killed,” said Mr Musk.
“And this person claimed asylum. And we let them in. A serial murderer. Who is proud of their murders. This is insane.”
In the last 10 days, law enforcement in Eagle Pass has arrested more than 17,000 illegal immigrants, according to the Washington Examiner.
The South African billionaire then claimed New York, AOC’s home state, was “buckling under the pressure” of illegal immigration.
“If New York can’t handle it, pretty much no part of the country can,” he said.
The congresswoman responded to Mr Musk’s live stream, which crashed after only four minutes, condemning his visit to the city and singling out his guide, local Republican congressman Tony Gonzalez.
“What’s funny about this photo?” she wrote on X on Thursday.
“The House is holding important votes in DC tonight, people are scrambling to avoid a shutdown, but this Republican Congressman decided to skip town to joyride with a billionaire when his own party has just a single-digit margin and needs his vote.”
Ms Ocasio-Cortez has long been critical of Mr Musk, and the pair’s spats frequently play out on Twitter.
She previously lashed out at Mr Musk online, writing: “Tired of having to collectively stress about what explosion of hate crimes is happening bc some billionaire with an ego problem unilaterally controls a massive communication platform.”
“Stop hitting on me, I’m really shy,” Mr Musk replied, adding a blushing emoji.
“I was talking about Zuckerberg, but ok,” she replied.
Ms Ocasio-Cortez also accused the billionaire of promoting a “sick” account impersonating her.
“FYI there’s a fake account on here impersonating me and going viral,” AOC tweeted. “The Twitter CEO has engaged it, boosting visibility.”
She claimed the account was “releasing false policy statements,” after previously tweeting about her boyfriend’s farts and her ideas to tackle climate change.
“I am assessing with my team how to move forward,” Ms Ocasio-Cortez said.
“In the meantime, be careful of what you see,” she warned her followers.
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