Xicheng district library can extend working hours without signing a contract. According to the provisions of the Labor Law of People's Republic of China (PRC), due to the needs of production and operation, the employer may extend the working hours after consultation with the trade unions and workers, and the daily working hours shall not exceed one hour; If it is necessary to extend the working hours for special reasons, the extended working hours shall not exceed three hours a day and thirty-six hours a month under the condition of ensuring the health of the workers. Therefore, it is possible for Xicheng District Library to extend its working hours without signing a contract.