From the Wikipedia page 
Battersea railway station (WLER) was a station on the West London Extension Railway on Battersea High Street. It opened on 1 Oct 1863 and closed on 21 October 1940 after air raid damage. It was south of Battersea railway bridge, at the end of the now pedestrianised area of Battersea High Street by the junction with Simpson Street. The station was demolished after closure leaving no remains today of the station.
It should not be confused with another station which Dendy Marshall (1963) lists as "Battersea"; but which Course (1963) calls "Battersea Park and Steamboat Pier station." This was located on the London, Brighton and South Coast Railway, adjacent to the steamboat pier on the south bank, next to Victoria railway bridge. It opened on 1 Oct 1860 with Victoria station, and closed on 1 November 1870.