ARIVACA, Ariz. — When the man walked in with fang marks on his leg, the volunteers knew the protocol: In the case of a rattlesnake bite, you call 911. But like all of the patients who end up here, his very presence in this desert clinic meant he had broken American law.

He’d been bitten sometime after he’d slipped across the southern border, while he was walking 12 miles through the arroyos and cactus-studded outcroppings in and around the Buenos Aires National Wildlife Refuge. He was hoping to reach his family in the United States; the last thing he wanted was for anyone to call the authorities.