hilbert space - Is non-linear quantum mechanics possible?

Say we have a state vector $|A\rangle$. Is it possible to have a theory where the evolution of $|A\rangle$ depends on the vector $|A\rangle$ itself? e.g.

$$i\frac{\partial}{\partial t} \psi(t) = \hat{F}(\psi(t)) \psi(t)$$

On reason I was thinking about this was the idea that space-time is related to entanglement. But entanglement means knowledge of the state-vector. So if this were true, the state-vector would have to affect itself.

In other words is non-linear quantum mechanics possible?