modular forms mod p and their liftings

Question

Deligne in "Courbes elliptique: Formulaire (d’après J. Tate)" computes the ring of modular forms mod $p$. He proves that if $p>3$, then the space of modular forms is isomorphic to $\mathbb{F}_p [c_4, c_6]$ but, for example, for $p=3$ one has $\mathbb{F}_3 [b_2, \Delta]$ where $b_2$ is of weight $2$ and $\Delta$ is of weight $12$. In particular, there is a nonzero weight $2$ modular form mod $3$. This modular form should not lift integrally and my question is: does it lift mod $9$? If yes, what is the minimal $k$ such that $b_2$ does not lift mod $3^k$?