I can - yes!
I am trained and have been approved by the Civil Aviation Authority. I fly in accordance with the Air Navigation Order (CAP393). Manned aviation pilots need to request permission to enter airspace near an airport and I am entitled to do the same. Getting that permission takes time and a lot of communication with the relevant parties, but it is possible.
There has been so much bad press about drones and airports - for example, the Gatwick incident - but that's only because that pilot did not have permission to be in the airspace. A trained and approved drone pilot can apply for these permissions.