Hmm, I must be hallucinating the fact that they advertise free WiFi as part of the first class offering...
https://www.britishairways.com/conte...-classes/first
Any statement that influences a consumer's purchasing decision becomes a term of the contract under section 50 of the Consumer Rights Act 2015.
BA's arrogance in believing that they can pick and choose which advertised benefits are 'part of the service offering' is pathetic. Of course it's part of the offering; it is literally a term of the contract that you will receive free WiFi. If they don't want it to be part of the offering they shouldn't advertise it.
Voting with your wallet is the best option going forwards, but I would not accept BA palming me off with a response like that. I would take it to CEDR if need be.