The Police: Hungry for You (J'aurais Toujours Faim De Toi). Written mostly in French, apparently with the help of Trudie Styler, with whom Sting was having an affair at the time.
Bill Wyman (Rolling Stones of course) - (Si Si) Je Suis un Rock Star (1981) - Has various parts in French, and looking at the quality of those parts it's very hard to believe Mr Wyman is fluent!
Je suis un rock star Je avais un residence Je habiter la A la south of France Voulez vous Partir with me? And come and rester la With me in France
In fact earlier in the song, he has a bash at Portuguese as well! Making this perhaps the only song written in TWO languages the author does not speak!
We danced to the music At the Mardi-Gras Then jumped on the Concorde You're so lah-di-dah Si, si Si, si Si, si
"Lah-di-dah" by the way is pure Cockney, so counts as three?! ;)
During research I found out he wrote it for Ian Dury - which makes perfect sense listening to it! He claims to have reluctantly recorded himself in "Cockney French" - Source
PS. Bonus points for "California Über Alles"? Just how many words does there have to be in this game?
This is an interesting question! The closest that I can offer is: Sanskrit. It's not really spoken any more in the sense of people carrying conversations in it (except in a unique village in India). It is, however, the language of many shlokas that are chanted even today in many households, so it's improper to call it a dead language. Though it doesn't fit the letter of your question, I believe it does fit the spirit.
There are many composers in the tradition of Indian classical music with compositions in Sanskrit. A tiny list of them follows:
As requested by the OP and with due respect to @Brahadeesh for making me think of it....
In the 1990's, British band Kula Shaker had a number of hits that were written with a substantial part of the lyrics in Sanskrit. Apparently, frontman Crispian Mills was somewhat obsessed with Indian culture.
Two of their 1996 Top 10 hits were called "Tattva" and "Govinder", which, for any non-English-speakers, are NOT English words!
Their album, "K" went to number 1 in the UK.