This British band had a U.K. #1 hit in 1964 with their debut single. It was also (probably) the first-ever #1 hit where the band's drummer was female. What's the name of the drummer?

This would be Honey Lantree of the band The Honeycombs. The big hit was 'Have I the Right?' and the female drummer garnered a lot of press attention. The record was produced by Joe Meek, who also wrote and recorded 'Telstar', recorded by the Tornados among other hits and is widely regarded as one of the most innovative producers of his era.

Honey's real name was Anne and she died in 2018.

https://www.allmusic.com/artist/the-honeycombs-mn0000765075 and https://www.allmusic.com/artist/honey-lantree-mn0001739486