This was on the Canadian embassy website: it describes the OP's scenario to a T:
Canadian-Argentine dual nationals may enter and leave Argentina on their Canadian passport only if their stay does not exceed 60 days. Once this period has elapsed, Argentine citizens must leave on a valid Argentine passport. The Argentine passport application process is lengthy, and the Embassy of Canada is not able to assist Canadian citizens in obtaining Argentine passports or other identity documents.
Canadian-Argentine dual nationals may enter and leave Argentina on their Canadian passport only if their stay does not exceed 60 days. Once this period has elapsed, Argentine citizens must leave on a valid Argentine passport. The Argentine passport application process is lengthy, and the Embassy of Canada is not able to assist Canadian citizens in obtaining Argentine passports or other identity documents.