UK citizen, resident in the Bahamas. Visa to visit Shanghai?

Yes, you can apply for an L type at the embassy.

Prepare the documents required for the application: your valid passport and a photocopy of its data page, an application form affixed with your recently-taken color photo, the original and photocopy of your residence card in Bahamas, and an invitation letter from your friend in Shanghai or roundtrip tickets and hotel reservations for China.