Do you mean the earliest recorded suicide of a person we can verify, or the earliest mention of someone committing suicide. If you mean the latter, then one that immediately comes to mind is the death of Ajax, mythical Greek hero of the Trojan War.

Considered the 2nd best warrior after Achilles, he becomes enraged after not being given a set of arnor crafted by the Gods, which was given to Odysseus. He is reportedly driven mad by Athena, and after coming to his senses is filled with grief and commits suicide with the sword given to him by Hector, Prince of Troy.

This story is written down as a play by Sophocles, which has survived to the present day. It was written around 440BCE. Additionally, Ajax has been depicted in sculpture and pottery found in the ancient Mediterranean, most notably the Belvedere Torso, which is believed depict Ajax as he contemplated his suicide.

Edit: I also know Ajax is referenced in The Odyssey when Odysseus travels to the underworld, but I don't recall if Iliad or Odyssey directly reference that Ajax killed himself. Both of those were written in the 8th century BCE, and are among the first Greek works written we still know of.

I suppose the men and women of Ancient Greece and Rome would be the place to look; these are civilisations with documentation that refer to actual people as well as the mythological.

The founding of the Roman Republic was the result of a suicide; that of Lucretia. She was the wife of an official of the last King of Rome, one of the Tarquins. This king raped her, and she killed herself rather than bring shame onto her family and husband. Supposedly this ignited the revolt that birthed the Republic. There are no contemporary sources, but several later sources (Livy is one) agree on the events, and that they occured around 500BC. The story is well known, and was well known at the time - she was frequently cited as a good role model for chaste young women and wives.