A basketball player jumps up to shoot a basket. Is momentum conserved?

(A) Yes, but only if you choose the correct system

(B) Yes, but only in the horizontal direction

(C) No, because the velocity of the basketball player changes with time

(D) It is a bad question, because momentum conservation is for objects moving at constant speed, and the basketball player is accelerating.

The correct option is (A) Yes, but only if you choose the correct system.
The momentum of a particle (p) is equal to the mass of the particle (m) times the velocity of particle (v).
So, p = mv …… (1)
From equation (1), p depends on v, the momentum (like the velocity) depends on the reference frame of the observer, and we must always specify this frame.
Here there is no external force acting on the system. Without any external force, the linear momentum of the system will be conserved. Momentum of the system is conserved if everything of the system is in elastic. So there is no loss of energy of the system. Therefore momentum is conserved but only if you choose the correct system.
From the above observation we conclude that, option (A) is correct.