In short yes - but then there are all the problem bits that people don't fully understand. To clear confusion here are some of them.
The Isles of Scilly are part of England, have representation in the Westminster Parliament (St Ives constituency) and are classed as a Unitary Authority - although not formally one, but for many local services are part of Cornwall and the South West.
Cornwall, despite a small but vocal nationalist grouping, has not been independent since 936 AD, when Athelstan finally invaded all of it.
All the Scottish Isles, Hebrides, Shetland etc are part of the Kingdom of Scotland and therefore part of the United Kingdom.
Attempts have been made to extend the United Kingdom - the small rock Rockall was claimed by the United Kingdom in 1955 when the navy landed by helicopter. This was to remove any claims by the Republic of Ireland, Denmark and Iceland. It was further made secure by an ex-SAS soldier, Tom MacLean, who camped on it for 40 days in 1975. This was to ensure the 12-mile exclusion zone, and to make sure its potential Oil/Gas reserves remained British. Noted explorer John Ridgway also lived there in the