What do the Argentineans call the islands known to the British and to the settlers living there as The Falkland Islands?

Las (Islas) Malvinas

-- from their French name Les Îles Malouines, bestowed by Admiral Louis Antoine de Bougainville in 1764 in commemoration of the first known settlers, who were mariners and fishermen from the Breton port of Saint-Malo in France.

The name Falkland Islands derives from that of the Falkland Channel which separates the two main islands. This name was bestowed by Captain John Strong, who landed on the islands in 1690, in honour of Anthony Cary, 5th Viscount of Falkland (a place in Scotland), who as Commissioner of the Admiralty had financed Strong's expedition.