It sounds like your partner may have an injury known as a mallet thumb. A mallet injury usually affects the distal interphalangeal or DIP joint of the finger. More rarely, the interphalangeal joint of the thumb is involved.
This joint is commonly injured during sporting activities such as baseball but evidently, calf roping is another way to end up with a mallet thumb. If the tip of the finger or thumb is struck enough force, the tendon that attaches to the small bone underneath can be injured. Untreated, this can cause the end of the thumb to fail to straighten completely, a condition called mallet thumb.
A review of the literature and current evidence reveals no consensus on the most effective or recommended treatment. And the evidence presented is based on published case reports. In general, closed injuries that do not require surgery are splinted. Open injuries are repaired surgically possibly using K-wire fixation of the interphalangeal joint of the thumb. When present, ruptured (avulsed) tendons of the thumb are reattached or reconstructed.
MRIs can be used to see if there is a tendon avulsion and how far the torn tendon has retracted (pulled away from the bone). The space between the end of the tendon and the place on the bone where it belongs is called a tendon gap. If the tendon hasn’t retracted too much, it can be stretched and pulled back to the insertion point and then reattached (repaired). If the tendon gap is too great, then a tendon graft may be needed (reconstruction).
It may be possible for your friend to avoid surgery with the use of a splint and some hand therapy. The most common time period for splint wear is four to six weeks when the injury is acute (occurred within the last two weeks). Treatment (and results) really depend on how much time has passed between injury and evaluation/treatment.
According to some studies, the earlier the treatment, the better the outcome. Closed injuries often results in better interphalangeal joint extension. Patients do not always get all of their normal motion (full thumb extension) back. A special (custom-made) thumb splint might be helpful but first, an examination by a hand surgeon or sports physician is really required. Encourage your roping partner to seek medical help and evaluation (preferrably sooner rather than later).