She has so far spent four nights in jail for contempt of court.
The judge in the case refuses to release her until she agrees to issue licences.
But the 49-year-old Apostolic Christian argued that should she not be made to carry out duties that go against her religious beliefs.
Madonna's brother, Christopher Ciccone, who is gay, has spoken out defending Kim Davis, writing on Facebook: "The county clerk deserves about as much support as you would give her if she were a Muslim (woman) who insisted on covering her face and refused not only gay marriage licences, but divorce, accusations of rape and driving a car without your man's approval."
He concluded: "Is it so difficult to allow this women her religion? Or must we destroy her in order for her to betray her faith.
"No matter how we judge, it's truth. The rights we have all fought for mean nothing if we deny her hers."
Judge David Bunning said that she will remain behind bars until she obeys the law.
Speaking to CNN he admitted that he was religious but that on becoming a judge the oath he took was above his personal beliefs, saying: "Her good faith belief is simply not a viable defence."
Five of her deputies agreed to issue marriage licenses in her absence on Thursday.
