LONDON, ENGLAND—Award-winning actor Mark Rylance launched a new campaign to save the Kawahiva—a small uncontacted hunter-gatherer tribe in Brazil’s Amazon rainforest. The Kawahiva are one of the most vulnerable peoples on the planet. Their forest is being invaded by armed loggers, miners and powerful ranchers in a region of Brazil’s Mato Grosso state known for its violence, rampant illegal logging and land grabs.

The uncontacted Indians are forced to live constantly on the run from invaders. Many of their relatives have been killed in genocidal attacks. The Kawahiva have demonstrated their wis...