Failure of dense-subspace extension into an incomplete codomain

Let $c_{00}$ have the ${\ell }^2$ norm, take $D=c_{00}\subset X={\ell }^2$, and let $T:D\to c_{00}$ be the identity. It is bounded but has no continuous extension ${\ell }^2\to c_{00}$, because continuity and density would force the extension to be the identity on every element of ${\ell }^2$.