Such a thing would not be possible. The departure of a Doctor and casting a new one are pretty big deals. There's no way for the BBC to practically stay silent about it. All it would take is for some tabloid to get wind of it, and prepare a report. BBC obligates itself not to lie to the public so they couldn't deny such a story. Indeed, the only legal way they can prevent the tabloid from running the story is to issue a public statement themselves effectively pre-empting the tabloid. This is the sort of thing that led to Eccleston's departure being announced before S1 finished airing, or Catherine Tate's return in S4 being announced before S3 finished airing. Rest assured, when the Doctor regenerates, it won't be a surprise. It will be known, it will be hyped. Which is how most people knew Tennant wouldn't really regenerate after The Stolen Earth.