No, a body can not have its velocity constant, while its speed varies. Rather, it can have its speed constant and its velocity varying. For example in a uniform circular motion.
The velocity of an object can be changed by:
- Change of speed
- Change of direction
And in a uniform circular motion, the direction is not always the same, rather, it keeps changing. While the speed can remain the same. So in this way, the velocity can vary and the speed can be constant. But the vice versa is never possible.