The CBP statement said the vehicle stopped several yards into Mexico.
The shooting occurred in the border city where US troops recently installed razor wire along portions of an 18-foot border wall, in some spots reaching almost ground level.
The council's resolution says the razor wire would harm or kill anyone who scales the wall and "is only found in a war, prison or battle setting" and shouldn't be in downtown Nogales.
The resolution demands the Trump administration remove the razor wire, but if the they fail to do so, then the city council of Nogales will file a lawsuit against the government for its removal.
City officials say Army troops installed more horizontal layers of the wire along the border wall last weekend.
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The CBP says the new wires are on USA government property.
USA 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.
According to U.S. Customs and Border Protection, a helicopter spotted the group amid freezing temperatures while investigating a possible campfire.
The mayor of Nogales, Arizona is threatening to sue to the federal government if it doesn't remove razor wire that now covers an entire stretch of border wall in the city.
Mexican authorities told the Nogales International newspaper that two men were in the truck and the passenger was arrested.
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.