Look up- Cork Ireland Please

[audbr]

Can any body help? I am looking for evidence of the birth of a Jane Ellen Audley in Cork. The date I have for her birth is 6th February 1859. This date is taken from the familysearch web site (IGI) which gives her date of birth as that above and gives her date of baptism as 1st January 1862 at St Peter Liverpool.
The information that she was born in Cork comes from the 1871 & 1881 census of Liverpool. I cannot find the family anywhere in the 1861 census of England and therefore presume that they must have been still in Ireland.

Is there any documentary evidence that she was born in Cork?