In the Wake of the Bounty was the second of the many films that have covered this subject.
Made in 1933 (two years before the more famous Charles Laughton version), it starred Mayne Lynton as William Bligh and Errol Flynn as Fletcher Christian. It was an Australian film written, directed and produced by Charles Chauvel. This was Flynn’s first movie and there are also rumours that he was descended from one of the Bounty mutineers (including Christian himself), although there does not appear to be any documentary evidence of this.
You can watch the film below.