JOHANNESBURG—A Russian merchant ship whose owner has allegedly carried weapons for the Kremlin turned off its transponder last month before surreptitiously docking at South Africa’s largest naval base, where it delivered and loaded unidentified cargoes, according to witnesses and a senior U.S. official.

South Africa has declined to say what the ship was carrying or what was loaded onto it at the Simon’s Town navy base. The country’s defense minister shrugged off U.S. concerns, saying Washington “threatens Africa, not just South Africa, of having anything that is even smelling of Russia.”