A Guatemalan migrant fell and was impaled on metal bars last week after she and her children tried to climb a wall on the United States border with Mexico, Customs and Border Protection said in a statement.The agency said that “pieces of rebar” had pierced the woman, a 26-year-old who has not been publicly identified, in the side of her torso and in her buttock on Friday about a mile from an official border crossing at San Ysidro, Calif. It released a picture on Sunday that showed her lying on her back in the dirt, her face out of the frame, with someone lifting her limp body to reveal a dark pool staining the ground beneath her.Rodney Scott, the chief patrol agent for the border agency’s San Diego sector, said the woman had risked her life climbing the border wall and could have died if not for emergency responders.“Entering our country illegally, particularly over our walls is not only dangerous, but also very foolish,” Mr. Scott said in the statement. “This woman placed her own life and her children’s lives in peril.”Many migrants trying to enter the United States are fleeing violence and poverty in Central America, but they have been seized on by President Trump, other Republicans and conservative news outlets as a national security threat.The woman was brought to a hospital, where she was treated for non-life-threatening injuries and her children were evaluated for “potential fall trauma,” the agency said. The children, ages 3 and 5, were later released into the custody of the Department of Homeland Security, which declined to say where they were being held on Tuesday.Border Patrol said its agents found the woman and her children on Friday about a mile east of the San Ysidro port of entry, the largest crossing between San Diego and Tijuana, Mexico, where scenes of chaos have erupted in recent days as members of a migrant caravan highlighted by Republicans as a midterm election issue have begun to arrive at the border.Despite the warnings from Republicans, the caravan appears to be made up largely of families, including women and children, fleeing unrest in their home countries. Thousands of migrants and asylum seekers have arrived at the border and become stuck in a processing bottleneck, with thousands more on the way.Theron Francisco, a Border Patrol spokesman, said the woman remained in the hospital on Tuesday.“Upon receiving medical clearance from a physician, she will be interviewed and processed at a nearby Border Patrol station,” he said. “Until then, her association with the migrant caravan and her ultimate disposition will remain unknown.”The San Ysidro crossing was closed for part of the day on Sunday after Customs and Border Protection officers fired tear gas into a crowd of hundreds of migrant protesters who were trying to evade a Mexican police cordon and approach the port of entry. It was reopened on Sunday night.The woman and her children were found in an area that Border Patrol described as “an active construction site.” The agency said 14 miles of new walls were being built there to replace barriers installed decades ago.