Let $\Omega$ be the completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the topology induced by the valuation $-\deg$. Does there exist a derivation on $\Omega$ that is a continuous extension of the classical derivation on $\mathbb F_q(T)$? If yes, is it unique?

Thanks in advance for any answer.