No. You do not have to to answer questions the police ask or give any type of a statement, other than giving your name and address.
Although you are not free to leave when you are arrested, you have the right to remain silent. This means you do not have to answer any questions asked by police (other than your name and address) or give the police any kind of statement, in writing or otherwise. The police may still ask you questions but you do not need to answer. You can say "I do not want to answer questions or give a statement". If you do say anything, it can be used against you in court.