Non-county "shires"
The suffix –shire could be a generalised term referring to a district. It did not acquire the strong association with county until later.
Other than these, the term was used for several other districts. Bedlingtonshire, Craikshire, Norhamshire and Islandshire were exclaves of County Durham, which were incorporated into Northumberland or Yorkshire in 1844. The suffix was also used for many hundreds, wapentakes and liberties such as Allertonshire, Blackburnshire, Halfshire, Howdenshire, Leylandshire, Powdershire, Pydarshire, Richmondshire, Riponshire, Salfordshire, Triggshire, Tynemouthshire, West Derbyshire and Wivelshire, counties corporate such as Hullshire, and other districts such as Applebyshire, Bamburghshire, Bunkleshire, Carlisleshire, Coldinghamshire, Coxwoldshire, Cravenshire, Hallamshire, Mashamshire and Yetholmshire.
Non-county shires were very common in Scotland. Kinross-shire and Clackmannanshire are arguably survivals from such districts. Non-county "shires" in Scotland include Bunkleshire, Coldinghamshire and Yetholmshire.
Richmondshire is today the name of a local government district of North Yorkshire.
Read more about this topic: Shire, Shire Names in The United Kingdom