# Continuous extension of the derivation in positive characteristic

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?