This depends on whether you mean the colloquial definition of a longsword, or the very specific definition of a cruciform "hand-and-a-half" sword that later evolved into the Zweihänder.

For the general definition, the Conquistadors who conquered much of the New World made extensive use of steel weapons. In the specific example of Pizarro's company that landed in Peru in 1531, they comprised 168 men, with 62 horses. The horses are noted to predominantly use the lance, but also the sword as a backup once they had charged. The footsoldiers predominantly used the sword. What specific type would have varied considerably, the Conquistadors were not a formalised army, they were effectively mercenaries given some leave by the Crown to conquer in exchange for a portion of the conquest. (The land obviously fell under the Crown, but prominent Conquistadors would become Encomenderos, basically small Lords of Estates.)

This meant that Conquistadors were entirely self-funded and equipped. It's likely that a high proportion of their swords would have been cruciform longswords, but the foot soldiers especially would have been equipped with swords as their primary weapon. Gunpowder weaponry was still scarce and in its infancy, they had only a handful of Harquebuses and two light cannons, if I remember correctly.

The steel longswords of the time were terrifyingly effective, however. Inca armour was mostly densely woven wool or cotton, with wooden plates on some combatants, and occasionally copper or bronze plates on high-ranking leaders. These were, of course, effectively paper to a steel longsword or lance. They were intended to prevent blunt force trauma from the predominantly blunt weapons used by the Inca and their contemporaries. Numerous accounts tell of Inca soldiers being hacked to pieces by the Spanish steel. The equipment difference was so stark that, in addition to steel armour and horses, the Spanish were able to slaughter thousands in exchange for one or two losses during the Siege of Cusco by Manco Inca. (In a battle involving tens or hundreds of thousands of natives against a few dozen Conquistadors, over the course of a year.)

So, to summarise, yes longswords as a general category were extensively used, though I cannot guarantee that they are the longsword.

References are to Last Days of the Incas by K. MacQuarrie. They use numerous accounts from Conquistadors to describe the Battle of Cajamarc in particular.