It was still being passed around by the beachgoers after its death and was later left discarded in the sand.
The dolphin was from the La Plata - also known as the Franciscana - variety, found off the coasts of Argentina, Uruguay and Brazil.
It is considered a "vulnerable" species because there are only about 30,000 left in the world.
The Argentine Wildlife Foundation later issued a reminder to the public of the mammal's extreme vulnerability, hoping it will prevent such a sad event again.
"The Franciscana, like other species, cannot remain for much time outside of the water, it has thick fatty skin which gives it heat and means that taking it out of the water rapidly causes it to dehydrate and die," it said in a statement.
"This occasion serves to inform the public about the urgent necessity to return these dolphins to the sea as soon as possible if they find them on the shore.
"It is fundamental that people help to rescue these animals, because every Franciscana counts now."