Are there non nilpotent operators with spectrum 0?

You'll have to go into infinite dimensional spaces. For example

$$V:=\left\{\,\{x_n\}_{n\in\Bbb N}\subset\Bbb R\right\}$$

with the usual operations of sum and scalar multiplcation, and the operator

$$R:V\to V\;\;,\;\;\;R\{x_1,.x_2,\ldots\}:=\{0,x_1,x_2,\ldots\}$$

has zero in its spectrum, but it is not nilpotent.

DonAntonio gave an algebraic example. If we put a norm on $V$, then the keyword is: quasinilpotent which, by definition means that the spectrum $=\{0\}$, ie the spectral radius $=0$. Note this is no longer algebraic since now invertible means bijective and bi-bounded.

This is equivalent to nilpotent in finite dimension (this follows readily from the Jordan normal form, with or without norm since boundedness is automatic in finite dimension), not in infinite dimension.

See the Volterra operator here for a counterexample. See also this related fact.