Is it possible to conserve the total kinetic energy of a system, but not its momentum?

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.

Suppose that energy is conserved in one frame of reference, and you want it to be conserved in all other frames as well. Conservation of momentum is exactly the condition you need in order to make this happen in all frames.

To see this, consider what happens when you change to a different frame of reference, $v\rightarrow v+u$. Then all kinetic energies transform according to $K\rightarrow K+muv+\text{const.}$ (Potential energies don't change under this transformation.)

Let's say we write your question as a conjecture: --

If energy is conserved and total KE is conserved, then momentum is conserved.

Then your conjecture can actually be strengthened to read: --

If energy is conserved, then momentum is conserved.

(This is implicitly assuming that we want all frames of reference to be valid.)

Is there any scenario which we could devise in which momentum is not conserved but kinetic energy is?

When a ball bounces off the ground or a wall. The momentum is flipped but the kinetic energy stays about the same.