This isn't really an answer. It's just to back up @GEdgar's chart showing that Argentine is overwhelmingly the standard form, but that Argentinian does occur.
I'm not going to look for "authoritative" sources. There's been debate about these two words for years, so obviously it's not clear-cut. Without checking, I expect there will be pedants who want Argentine restricted to the silver-like meaning, and historians who point out that we used to call the country itself The Argentine, so the inhabitants must be called something else.
I think what happens is most of us rarely need either word, so we don't really know or care which is "correct". As shown by this NGram, Brits (but not Americans) started using Argentinian more after WW2 ended (we imported lots of beef), and after the Falklands war (which we all talked about).
After each of those "peaks", the alternative non-standard form gradually faded as the people using it realised they were in a minority. But because educated people never use Argentinian for the silvery meaning, there's are always some people who want to keep that for the nationality. But there are never enough people thinking like that, so it fades away again.