Which foreign soldier vowed to follow David "whether it means life or death"?

Answer: Ittai the Gittite

David urged Ittai, a foreigner and exile, to go back, but Ittai swore to stay wherever the king was, and he marched on with David.

“Then the king said to Ittai the Gittite, “Why should you also go with us? Go back and stay with the new king, since you are both a foreigner and an exile from your homeland.”— 2 Samuel 15:19 (BSB)