Turning to the experts ...
The Associated Press stylebook, which is what most American news sources use to settle most style questions, goes with .... wait for it ... Argentine, not Argentinian.
This is the 2015 edition:
"Argentine The preferred term for the people and culture of Argentina."
That is somewhat different from the Merriam-Webster dictionary, which first lists Argentine, but also appears to accept Argentinean or Argentinian.
They also note that argentine with a lowercase "a" can mean "silvery" as an adjective or just "silver" as a noun.
This dictionary, as well as Wikipedia, also says that the official name of the nation in English is The Argentine Republic. (Spanish: República Argentina)
Dictionary.com goes with Argentine or Argentinean and doesn't even list Argentinian.