“O.K., Mommy.”

The truth is that Ms. de la Cruz does not know when her daughter will be released.

They live in the border town of Laredo illegally, and when Ms. de la Cruz sent Rosa Maria to Corpus Christi for the surgery early Tuesday, the girl’s ambulance was stopped at a Border Patrol checkpoint. After agents learned the girl was an undocumented immigrant, they followed her to the hospital and guarded her room. When she was discharged, the agents took her to the shelter, which houses juveniles in immigration custody.

“It’s the period when she needs me the most,” said Ms. de la Cruz, who cannot visit her daughter because she, too, could be arrested at a checkpoint. “I can’t help her. When I start to think about her, I start to get sad.”

In a year when President Trump’s hard line on illegal immigration has driven the number of immigration arrests up by more than 40 percent, Rosa Maria’s case has sped straight to the heart of the immigration-debate maelstrom.

Politicians have called for her release; activists have rallied, fund-raised and prayed on her behalf, questioning whether the Border Patrol violated its own policy against arresting immigrants at hospitals and why agents chose to expend so much time and manpower on a disabled 10-year-old girl.

The Border Patrol said it had followed proper procedure in Rosa Maria’s case. And to supporters of tougher enforcement, Rosa Maria and her family — no matter how sympathetic their situation — are the embodiment of the argument that a weak border only encourages immigrants to come to the United States and take advantage of its schools, health care and other resources.