Thank you lancsann.
I hadn't found that. I had searched by the individual's name and by Lancaster Asylum - but no patients records showed up.
So it looks like it WAS Lancaster Moor, as Rosie had first thought. This extract from the Archives is really interesting;
"Background information
Lancaster Moor Hospital was Lancashire's first County Lunatic Asylum. The decision to build it was taken in 1809, one year after the permissive County Asylums Act, 1808. The hospital opened in 1816 as the "County Lunatic Asylum for the County Palatine of Lancaster". It was only the fourth asylum to be built under the terms of the Act in the country. It was extended in 1824 and 1883, and by 1891 it accommodated 1833 patients. In that year its administration was transferred to the new Lancashire Asylums Board of Lancashire County Council. Additional buildings, known as Ridge Lea, on the "villa" principle were added in 1907, 1909, 1912, 1916 and 1938. These buildings were chiefly to accommodate private patients."
By coincidence it mentions the year 1891 - this was the year the person I am seeking was admitted.
So it had been in existence since 1816!!