Copies of the newspaper advert and the flag were torched at a cruise terminal in the capital after British Prime Minister David Cameron rejected calls for talks on the disputed territory.
A group of the islanders, called Falklands United, responded on Friday to a letter by Cristina Fernandez de Kirchner published in The Guardian and Independent in which she called for the UK to give up the Falklands.
They wrote: "Our home is a British Overseas Territory, not a colony as you seemingly wish to convince people.
"We have never been prouder of our association with the United Kingdom and our unique relationship.
"Any decision to change that would be OUR and not YOUR choice.