After a disco period, Jamaican model and singer Grace Jones brought out a dub-heavy reggae album of covers in 1980, and had a hit single with her cover of The Pretenders song Private Life. Arguably even more astonishing is her take on Joy Division's She's Lost Control, which formed the B-side of the single.
Early punk had a large crossover with reggae music - stemming largely from DJs like Don Letts, whose club nights in Brixton found an audience appreciative of both genres.
Many punk bands showed reggae influences, The Ruts and Stiff Little Fingers would play originals and/or covers from the genre, while PiL's second album Metal Box was heavily influenced by dub. The Clash would play plenty of reggae too, and the one time I saw them live they even had Mikey Dredd supporting them on tour.
Chrissie Hynde of The Pretenders was also in on this scene, and their 1980 album Pretenders included her own composition Private Life in a Jamaican style. Very soon after that album was released, Grace Jones was recording with dub and reggae legends Sly & Robbie, and included a cover of Private Life on her own LP. It was a hit and still sounds amazing to me now.
Perhaps less well known was the non-album B-side, a cover of Joy Division's She's Lost Control. The full 12" version of this is an amazing piece of dub and I wholeheartedly recommend it.
Even as an avid Joy Division fan, I suggest that both sides of this single can, at the very least, be considered for the rare title of "cover better than the original".