The authors define a subspace $L$ to be weak$^*$-extensible in $X$ if every weak$^*$-null sequence of $L^*$ admits a subsequence that can be extended to a weak$^*$-null sequence of $X^*$. Also, a Banach space $X$ is said to be weak$^*$-extensible if all of its subspaces are weak$^*$-extensible in $X$. This paper contains a basic study of weak$^*$-extensible Banach spaces. The authors observe that if the unit ball of $X$ is weak$^*$-sequentially complete then $X$ is weak$^*$-extensible, that the product of a weak$^*$-extensible by a finite-dimensional space is weak$^*$-extensible, and that $c_0$ is not weak$^*$-extensible in $\ell_\infty$. More interesting are the two questions after Remark 4.7: Does every weak$^*$-extensible space have a weak$^*$-sequentially compact dual ball? Is being weak$^*$-extensible a $3$-space property? \par The second question has a negative answer and the counterexample is the same that works as a counterexample for the $3$-space problem for having a weak$^*$-sequentially compact dual ball (and which the authors mention in passing). Manuel González and Anatolij Plichko have also observed this.