There is a story that the Sri Lankans hold dear. It goes back to somewhere around the 6th BCE when a banished prince, the grandson of a princess of Kalinga (who had eloped with a lion), made his way to the island deep south with a band of 700 men. He defeated the local inhabitants the yakkhas and went on to rule Sri Lanka as King Vijaya. By now, he had taken the title of his father and grandfather (the lion) and that is how his people came to be known as the Sinhalese.
This story is mentioned in the Buddhist text Mahavamsa and the earlier Deepavamsa which chronicle the history of the people of Sri Lanka. They were both written somewhere between the 3rd and 5th CE – roughly 900 years later.