Mexico is the world's largest Silver producing country followed by Peru and Australia, however the Cannington Silver mine near Mount Isa in Queensland Australia is the world's largest single producer of Silver
Historically it would be Potosi in Bolivia, which between 16th-early 19th cennturies produced gazillions worth of silver until the mines were exhausted. In the 16th century the area was thje world's largest industrial complex -- based on silver extraction.