ANCUD, Chile—On a recent day in this remote island town, Dr. Jhomar Yansen rushed to operate on a man who had been kicked in the gut by a sheep and was suffering severe internal bleeding.

The surgeon worked in his native Venezuela saving patients with gunshot and knife wounds until last year, when he fled the chaotic, impoverished country like thousands of other professionals. Now he finds himself as part of a cluster of Venezuelan physicians looking after fishermen and shepherds here on a green archipelago in the frigid Southern...