A body can have

The fact that a body has zero momentum denotes that it either has no mass or no velocity. The kinetic energy ($\mathrm{K}=\frac{1}{2}{\mathrm{mv}}^2$) will be zero if either the mass or the velocity are both zero. ( example: a vertically projected body at its highest point)

But a body cannot have kinetic energy without also possessing momentum.