On the way to the Jordan, who kept telling Elijah, "As surely as the LORD lives... I will not leave you"?

Answer: Elisha

At Gilgal, Bethel and Jericho Elijah urged Elisha to stay behind, but each time Elisha refused to leave him.

“and Elijah said to Elisha, “Please stay here, for the LORD has sent me on to Bethel.” But Elisha replied, “As surely as the LORD lives and as you yourself live, I will not leave you.” So they went down to Bethel.”— 2 Kings 2:2 (BSB)