Yeah, but it's not a county. It's a... something. I think it has a borough council? District council?
As you may know from your inlaws, when most of the peninsula was assigned to Merseyside, even the bits in Cheshire got stuck with 'L' Liverpool postcodes, which obviously went down very badly with the posho golf and pony folk in Caldy and the like. But after years of very British moaning, on the Cheshire side of the border they were reassigned to 'CH' for Chester, and doubtless celebrated with some scones and jam at the Women's Institute.
In Caldy they probably pretend their 'L' stands for 'Lord of the Manor'.