Movie studios are jostling to be the first to bring the story of WikiLeaks and its founder, Julian Assange, to the big screen. But when Hollywood attempts to harness a swift-moving news story—particularly one lacking a finale—the task can be more complicated than simply casting a look-alike.

As the famously mercurial Mr. Assange battles his way through protracted legal challenges, Hollywood studio executives have become entranced by the sweep, stakes and unpredictability of his story.

The...