Ramatam has April Lawton on guitar, but no vocals.
There are questions as to whether she was actually born female, but she presented as female during her career.
White Zombie
The Cramps
Motley Crue had a female drummer for a few years while Tommy Lee was out of the band.
There’s one well-known rock group out there that consists of four old men (and no women) that don’t even sing.
Mount Rushmore
Electric Light Orchestra’s touring line-up for the past few years has featured a three-piece string section: violinist Rosie Langley, and cellists Amy Langley and Jess Cox.
However, it doesn’t appear that Jeff Lynne used strings, or that all-female string section, on his last two ELO studio albums (which are, effectively, solo albums, as Lynne played nearly all of the instruments, as well as performing the vocals).
Zappa/Mothers - Ruth Underwood
Pulp (biggest hits included Common People) had Candida Doyle on keyboards. Jarvis Cocker was lead singer.
TCMF-2L
New Order (one of their biggest hits Blue Monday) have Gillian Gilbert on keyboards. Bernard Sumner on lead vocals.
TCMF-2L
Jayhawks and Drive-By Truckers (well, up until Shona left)
Trip Shakespeare: drummer Elaine Harris was the only member who did not sing lead.
Well, there’s the E Street Band, which in its current incarnation has two women, neither of which sing lead.
Incendio has a female rhythm section. No vocalists.
I’m assuming the OP is discounting every all-female group, since one of them would be singing lead sometime. I would also assume that strictly back-up singers wouldn’t count either.
The Rasmus, a rock quartet out of Finland, has men on drums, bass, and lead vocals, and a woman on lead guitar.
They represented their homeland in the Eurovision Song Contest last month. Here’s the song, “Jezebel.”
(It might be geo-locked for US viewers. If so, search the title to see if there’s a regionally agnostic alternative.)
Camper Van Beethoven had Morgan Fichter on violin for the Key Lime Pie album and tour.
The Waterboys have had something like 85 members over the years counting touring members (they’re very Spinal Tap-esque on that front). Sharon Shannon on accordion was a notable studio addition on the Room to Roam album and tour.
And Sex-Bob-Omb had a girl drummer…in fact every band in the Scott Pilgrim movie has a girl drummer, kind of a running joke.
I frequently watch new groups when they appear on the late night shows. Most of them have both male and female members, and sometimes the lead singer is the male. I couldn’t put a name on them if my life depended on it, but they seem to be much more common today that 50 years ago.
Having a women play an instrument but not sing lead goes back, like so much else, to Bo Diddley. Peggy Jones was his rhythm guitarist for several years.
Patricia Morrison (Sisters of Mercy and others) played with The Damned for a few years before starting a family with David Vanian.
I was going to say that it must be pretty common, but I think that it’s probably just a coincidence that my favorite bands have a female playing an instrument with a male vocalist. Smashing Pumpkins are my all-time favorite band ever, but White Stripes and Silversun Pickups are possibly my second favorites. I also love the Pixies and Velvet Underground.
I did not realize that Talking Heads had a female bassist. Their videos mostly show David Byrne, I honestly couldn’t even name another member of that band.
Their similarity to SP was what drew me to them in the first place. I also think Jack White sings a bit like Billy Corgan. That’s probably why they are my second-favorite bands. 
ETA: The New Radicals (which only released one album, admittedly, and was a one-hit wonder with “You Get What You Give”) had a main singer in Gregg Alexander and Danielle Brisebois was a backup singer/keyboardist/percussionist. The rest of the band was a rotating cast.
Jimmy Buffet and the Coral Reefers, has two female singers, and one gets to occasionally sing solo.
Tina Weymouth, who has also been married to the Talking Heads’ drummer, Chris Frantz, since 1977.
KMFDM
Sascha Konietzko is the lead singer for sure, but Lucia Cifarelli does do a fair amount of singing in the band. If you only followed KMFDM back in the 80’s and 90’s, you would not even realize they have now a woman in the band for 20 years. They do and she sings lead on about 2 songs per album, but is keyboards and backing vocals mostly.
A Perfect Circle when Paz Lenchantin was in the band.