We present a design of an 8-bit bit-serial rapid single-flux-quantum (RSFQ) microprocessor, which is called CORE e4, and its high-speed functionality test results. The CORE e4 is equipped with four general-purpose registers and can execute 20 different instructions. The first version of the CORE e4, which is called CORE e4v1, has been implemented using the National Institute of Advanced Industrial Science and Technology (AIST) 10-kA/cm2 advanced process (ADP2) and the CONNECT cell library. The CORE e4v1 consists of approximately 7000 Josephson junctions and occupies a circuit area of 3.00 mm $\times$ 1.98 mm. The estimated power consumption and performance are 2.03 mW and 333 million instructions per second, respectively. CORE e4 is one of the highest performing RSFQ microprocessors.