1. **Stating the problem:** We are given that $y \neq 0$, $x = \frac{4}{y}$, and $\frac{y}{z} = 8$. We need to find the value of $\frac{z}{y}$.\n\n2. **Understanding the given information:** The equation $\frac{y}{z} = 8$ means that $y$ is 8 times $z$. We want to find the reciprocal ratio $\frac{z}{y}$.\n\n3. **Using the reciprocal property:** If $\frac{y}{z} = 8$, then by taking the reciprocal of both sides, we get $$\frac{z}{y} = \frac{1}{8}.$$\n\n4. **Conclusion:** Therefore, the value of $\frac{z}{y}$ is $\frac{1}{8}$.