On Tuesday, DHS pushed back against what it called "false narratives" surrounding what had happened, blaming sanctuary cities and media for "spreading lies" and discouraging people from resisting officers and evading arrest.

"Contrary to their claims, this person is an illegal alien who has no legal status to be here and has been given some of the best medical care he's ever received in his life," said DHS Secretary Markwayne Mullin in a statement to the BBC.

DHS said the man, whom they did not name, was "the subject of a targeted vehicle stop" and that his work permit expired last year.

"No amount of smears and false narratives will slow down ICE from doing its job by removing dangerous illegal aliens from our communities and making America safe again," Mullin said.

The Austin Police Department, which was not involved in the incident, released dashcam and body camera footage on Tuesday showing its officers at the scene.

It does not show the shooting, but several gunshots can be heard in one dashcam clip, while another shows officers approaching Garces Perez's car and detaining him. At one point, Garces Perez's t-shirt is cut to reveal what appears to be a single gunshot in his back.

Lincoln-Goldfinch previously told the BBC that after being unable to move his arm in the detention facility, Garces Perez was taken back to hospital on Monday evening. He was treated there and has since been returned to the detention facility.

He said he was handcuffed, shackled at the ankles, placed in a waist chain, and handled roughly during his transfer, despite his injury, she said.

Lincoln-Goldfinch added that detention personnel had revoked his communication privileges and moved him from an infirmary into an isolated room with a flat, non-adjustable bed.

In its statement on Tuesday, DHS said Garces Perez had received medical care "around the clock" and denied claims he received no pain medication and was forced to sleep on the floor.

"This alien was released from the hospital and arrived at the detention facility just a few hours later, where he slept on a bed," DHS said.