Is a Hahn-Banach extension always continuous?

The formulation of Hahn-Banach that you have does not require any topology on $X$. One obvious application is the case where $X$ does have a norm and you use the theorem to extend functionals while preserving the norm.

But the form with the seminorm $q$ is the one that allows one to obtain the geometric form of Hahn-Banach (i.e., the separation theorems) which is often the most useful.

Here is a counterexample: Let $X$ be an infinite-dimensional normed vector space. Then, there is a discontinuous linear functional $q$. Evidently, $q$ is sublinear. Now, take $U = \{0\}$ and $l(0) = 0$. However, the only linear function $L : X \to \mathbb R$ satisfying $q$ is $L = q$ which is discontinuous.