Which is the only ungulate (hoofed animal) whose lower incisors grow constantly like a rodent’s ones?

The Vicuña.

https://southwickszoo.com/our-animals/vicuna/

The reason why vicuñas have continuously growing lower incisors is the protection of their habitat: the Andean altiplano and its natural mountain pastures. While sheep or goats uproot the plants they eat, which causes the land to crumble and turn to dust, which then blows away, the vicuña cuts them with its lower incisors that grow continually and are covered with hard enamel.

By the way: this is not the only adaptation to protect the pastures: Unlike sheep hooves, which cut the hard soil and contribute to erosion, the foot of the vicuña is covered with soft pads, which allow it to hold fast on the rocky surfaces and which prevent damage to the soil.

Source: CERES No. 116 (Vol. 29, No. 2) - March-April 1987