**Step 1 — Write the distance formula.**
$$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$

**Step 2 — Substitute the given points and distance.**
$$5 = \sqrt{(1-4)^2 + (0-p)^2}$$

**Step 3 — Simplify inside the root.**
$$5 = \sqrt{9 + p^2}$$

**Step 4 — Square both sides.**
$$25 = 9 + p^2 \ \Rightarrow\ p^2 = 16$$

**Step 5 — Take the square root.**
$$p = \pm 4$$

$$\boxed{p = 4 \ \text{ or } \ p = -4}$$