I think the Hahn-Banach extension theorem states:
If $X$ is a Hausdorff lcs, $M\subseteq X$ a subspace and $f:M\to\Bbb K$ is a continuous linear map, then there is a continuous linear map $F:X\to\Bbb K$ with $F\lvert_M=f$.
However I cannot find a reference for this version. Usually the Hahn-Banach extension theorem is states that a functional dominated by one sub-linear function can have its domain extended so that the domination remains intact.
In the case of a locally convex space one usually has an infinite amount of semi-norms generating the topology. Suppose one has $C_p\in\Bbb R$ so that $|f(x)|≤ C_p p(x)$ for all semi-norms $p$ and $x\in M$, can one choose an extension so that this system of inequalities remains true for all $x\in X$?