Yes, it is possible to design such a rectangular park. The length and breadth are both 20 m.

Let the length be $l$ m and breadth be $b$ m.

Given: Perimeter = $80$ m and Area = $400$ m²

From perimeter:

$$2(l + b) = 80$$

Therefore, $l + b = 40$, which gives $l = 40 - b$ ... (1)

From area:

$$l \times b = 400 \quad \text{... (2)}$$

Substituting (1) in (2):

$$(40 - b) \times b = 400$$ $$40b - b^2 = 400$$ $$b^2 - 40b + 400 = 0$$

To check if real roots exist, we find the discriminant:

$$b^2 - 4ac = (-40)^2 - 4(1)(400) = 1600 - 1600 = 0$$

Since the discriminant equals 0, the quadratic equation has two equal real roots. Therefore, it is possible to design such a park.

Solving $b^2 - 40b + 400 = 0$:

$$(b - 20)^2 = 0$$ $$b = 20 \text{ m}$$

From (1): $l = 40 - 20 = 20$ m

Verification: Perimeter = $2(20 + 20) = 80$ m and Area = $20 \times 20 = 400$ m²

Thus, the park is a square with both length and breadth equal to 20 m.