Question

is there is any situation where a body has a constant speed but variable velocity

Answer 1

Yes, example of such situation is a satellite orbiting the earth in a circular motion. The speed is constant, whereas since the path is circular, it's direction is variable which means it's velocity is also variable.

Answer 2

Yes, it can happen.

Consider an object with constant speed moving along a circular path

Here, at each point, direction of velocity is different. So velocity is not constant. However the speed is constant