In 1 Samuel 21:10, David "flees" to Gath, and King Achish's servants say to him, "Is this not David, the king of the land?"
This is positioned immediately after David's goodbye to Jonathan (1 Samuel 20) and retrieval of Goliath's sword from the priests at Nob (1 Samuel 21:1-9), so well before Saul's death and the presumed beginning of David's kingship.
There's a thought David might've seized Hebron before Saul died, based largely on David reigning 5.5 years longer in Hebron than Ishbosheth did in the North (meaning he either took 5.5 years to move his base from Hebron to Jerusalem after Ishbosheth's death or he was already reigning in Hebron before Saul died and Ishbosheth took the throne), but this article also makes a good point about David going to Keilah's rescue -- Keilah was a southern city, so if David were already king of the south at this point, it makes sense for David to have gone to their aid and not Saul.
But even supposing David's reign in Hebron began before Saul's death, why would the servants of Gath already know David as "king of the land" in 1 Samuel 21? If he were already king of Hebron here, why would he be "fleeing" to Gath? Is it possible these verses reflect a tradition of a later event that had no part in David's fleeing?