Suppose that a linear system of inequalities $Ax \le b$, where $A\in Z^{m\times n}$ and $b\in Z^m$ have integral coefficients, has an infinite number of integral solutions $x$.

Can one conclude that there is a ray $\{x+tp\mid t\ge 0\}$ containing infinitely many integral solutions? (If one drops the integrality condition, the answer is clearly yes.)