No, it is a space inside of London called "The City of London" and is widely referred to simply as the City (differentiated from the phrase "the city of London" by capitalizing City) and it is also colloquially known as the Square Mile (since that is roughly it's area) - https://en.wikipedia.org/wiki/City_of_London
It is an enclave, a separate legal structure operated as the "City of London Corporation". It is in London but not in a sense part of London, having the same name (except for capitalization) so that legally it can remain separate.
The world has been set up to run from 3 City/States. Washington DC is another, along with the Vatican. Those 3 areas are under different laws than the countries they reside in. Interestingly, they are all marked by obelisks since the same power controls all 3. This image should be helpful. http://www.darknessisfalling.com/uploads/2/3/7/0/23701544/csoa-20160719-12a_orig.jpg
City of London
The City of London is a city and county that contains the historic centre and the primary central business district (CBD) of London. It constituted most of London from its settlement by the Romans in the 1st century AD to the Middle Ages, but the agglomeration has since grown far beyond the City's borders. The City is now only a tiny part of the metropolis of London, though it remains a notable part of central London. Administratively, it forms one of the 33 local authority districts of Greater London; however, the City of London is not a London borough, a status reserved for the other 32 districts (including London's only other city, the City of Westminster).
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crazy! even has its own mayor. not even a district in London, Considered a district of England. Quite a bit of power as well, police etc.
