Let $B_t$ be a standard one dimensional Brownian motion and $L_t$ its local time at zero. Setting $\beta_t=\int_0^t\text{sgn}(B_s)dB_s$, it is well-known that $$L_t=|B_t|-\beta_t$$ Now defining $X_t=\beta_t+L_t$, we obviously have that $X_t=|B_t|$, in particular $X_t$ is a non-negative process.
If I now applay the generalized Ito formula to the process $|X_t|$ I obtain the following contradiction:
$$|X_t|=X_t+1/2 L_t$$
and so $L_t$ is the zero process.
Where is the mistake?