Jensen's
The Mamluk Lancer unfortunately doesn't translate the lance exercises. However, the bibliography looks promising. I have "La parure des cavaliers..." at hand, and while it doesn't specifically mention tent-pegging (the section specifically devoted to the lance only mentions tilting at rings), Appendix VI makes it sound like it, or something very similar, should be relatively easy to find in muslim sources:
p397 wrote:J'en retiens que les jeux de la bague, des anneaux posés par terre, en guise de têtes, des cibles de toute sorte, etc., étaient pratiqués couramment par les cavaliers réguliers des cours musulmanes
As for the ancient world: Arrian mentions that part of the Hippika Gymnasia (albeit with swords) was to imitate the action of killing a fallen enemy; if this were done with lance instead of sword you would get tent-pegging.