A 1.44 square mile biological Site of Special Scientific Interest in Buckinghamshire. The southern half is owned by the Corporation of London and is open to the public. Its area is slightly larger than the 'Square Mile' (actually 1.12 square miles) itself.
From Facebook: Located in South Buckinghamshire and around 25 miles from London, Burnham Beeches was acquired by the City of London in 1880, in response to a threatened purchase by residential developers. It was for sale as “land suitable for the erection of superior residences”.
Londonist have addressed the issue of why the City of London owns large areas of green space outside its own Square Mile, such as Epping Forest, and some, such as Burnham Beeches, entirely outside London.
But nowhere can we find an explanation of why the City bought Burnham Beeches.
