In order for momentum to be conserved, it must be the case that $$\mathbf F_\text{net}=\frac{\text d\mathbf p}{\text dt}=0$$

In order for kinetic energy to be conserved, it must be the case that $$\text dK=\text dW_\text{net}=\mathbf F_\text{net}\cdot\text d\mathbf x=0$$ at all instants in time.

So, is there a case where the net work done on an object is $0$, yet there is still a net force acting on the object? The answer is yes! If $\mathbf F_\text{net}$ is perpendicular to the path of the object at all times. A simple example of this is an object undergoing uniform circular motion. The object's kinetic energy is not changing (as its speed remains constant), yet the momentum is constantly changing due to the non-zero net force.

In order for momentum to be conserved, it must be the case that $$\mathbf F_\text{net}=\frac{\text d\mathbf p}{\text dt}=0$$

In order for kinetic energy to be conserved, it must be the case that $$\text dK=\text dW_\text{net}=\mathbf F_\text{net}\cdot\text d\mathbf x=0$$ at all instants in time.

So, is there a case where the net work done on an object is $0$, yet there is still a net force acting on the object? The answer is yes! We just need $\mathbf F_\text{net}\neq0$ to be perpendicular to the path of the object at all times. A simple example of this is an object undergoing uniform circular motion. The object's kinetic energy is not changing (as its speed remains constant), yet the momentum is constantly changing due to the non-zero net force.