Problem 3

When the length of the equal sides is $3$, the lengths of the three sides of the triangle are: $3$, $3$, $8$;
Since $3 + 3 < 8$, it cannot form a triangle;
Therefore, the length of the equal sides of this isosceles triangle must be $8$, and its perimeter is $8 + 8 + 3 = 19$.
Hence, the correct answer is $\boxed{B}$.
This problem should first determine the lengths of the equal sides and the base of the isosceles triangle based on the relationship between the sides of a triangle, and then calculate the perimeter of the triangle.
This question examines the properties of isosceles triangles and the relationship between the sides of a triangle; for problems where the equal sides and base are not explicitly known, it is essential to consider both possibilities, discuss them separately, and verify whether each scenario can form a triangle. This is a crucial point and the key to solving the problem.