The mayor of Nogales, Arizona said the officer opened fire after the driver of a truck came barreling toward him and refused to stop.
According to CBP, the pickup crossed a few yards into Mexico before it struck a concrete barrier.
The resolution called the razor wire "not only irresponsible, but inhumane" and argues that this type of wall "is only found in a war, prison or battle setting".
Garino told CBS Evening News last month that adding to the border wall, which his city has had for more than 20 years, would be ineffective.
Officers referred the driver of a Ford sedan, later identified as a 29-year-old Mexican male for additional inspection as he attempted to enter the USA early Thursday morning.
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US authorities arrested or stopped people for immigration violations 58,207 times in January, down 4 percent from 60,779 in December but up 62 percent from 35,905 in January 2018.
As tensions are heating up in Nogales over concertina wire strung up across the border fence, the mayor of Douglas, Arizona, says increased border militarization is making life worse for residents in his city.
CBP said it would release more information about the shooting.
Nogales, a city of about 20,000 people, is largely reliant on Mexican shoppers and cross-border trade. "The shooting occurred at the Port of Nogales, DeConcini Crossing in Nogales, Arizona".
Overall, illegal crossings at the southern border have dropped dramatically compared to previous decades but in recent years the number of families and unaccompanied children heading to the United States has increased.