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?
 A: The superposition principle, which is basically equivalent to a linear evolution equation of the quantum state, is very fundamental for quantum mechanics.
As eranreches mentioned in his comment, however, effective equations for many particles are often non-linear, e.g. the Gross-Pitaevskii equation or the Hartree-Fock equation. They feature an interaction term of the form $V * |\psi|^2$, where $*$ denotes the convolution. But there, the basic superposition principle is not changed and only the effective descriptions yields the non-linearity.
There is also a formulation of quantum mechanics in which the collapse is effectively modelled in the Schrödinger equation itself. This leads to a nonlinear, stochastic equation. These models are called collapse models. The first one was by GRW (Ghirardi, Rimini, Weber, see here) in  and there are newer ones under the keyword CSL (continuous spontaneous localization). If such a theory is true - and they are not fully falsified yet - the superposition principle is violated a tiny little bit.
A: I assume you mean nonlinearity in quantum mechanics at a fundamental level, therefore I do not go into what others have said about `emergent nonlinearity', for example as in  Gross-Pitaevskii equation.
It would be superficial to make this question a matter of definition: indeed there is a certainly defined theory called quantum mechanics and that theory by definition is not nonlinear! But this is obviously not meant by the question.
A direct answer to your question is yes, of course. There is the Doebner-Goldin equation
$$
i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2 \psi + V\psi + iD\hbar\left(\frac{|\nabla\psi|^2}{|\psi|^2}\psi+\nabla^2 \psi \right)
$$
with the same form as what you mentioned in the question. The nonlinearity of Doebner-Goldin is assumed to be fundamental.
To think about foundations of quantum mechanics (more generally, any physical theory) we must not forget the logical-historical order/structure of the theory: some might answer this question using Hilbert spaces, or more, try to approach nonlinearity in quantum mechanics using Hilbert spaces. But we must not forget that the primary element of any quantum mechanics must be a wave equation governing the evolution of de Broglie waves. Hilbert spaces, Born rule, etc. come only after that. In other words, it is the Schrödinger equation that leads us to the whole notion of Hilbert space and the consequent orthodox formalism, not the other way around.
Therefore a nonlinear generalization of Schrödinger equation might demand revision of the whole notion of Hilbert space, state vectors (implying that your question is implicitly assuming some linear structure by mere use of the word `vector'), etc.
I have investigated these issues in detail in my paper.
A: 
Is non-linear quantum mechanics possible?

No ─ quantum mechanics is at its very heart a linear theory; that's the core of what some like to call "the wave nature of matter" and what really distinguishes it from classical mechanics. 
Of course, it's possible that there's a deeper theory that underlies QM which includes that type of nonlinearity and which only reduces to QM in some suitable limit (like relativity reduces to newtonian mechanics) and that we've yet to find the way out of that limit. We know that that could have very striking consequences and it's not impossible ─ but if it's true, we wouldn't call that theory QM.
All of that said, though:

Is a non-linear quantum-mechanics-like formalism possible?

Yes, absolutely. The simplest such version is the so-called non-linear Schrödinger equation, and its close cousin the Gross-Pitaevskii equation,
$$
i\hbar \frac{\partial}{\partial t} \psi(\mathbf r,t) = \left[ -\frac12 \nabla^2 + V(\mathbf r) + g |\psi(\mathbf r,t)|^2 \right] \psi(\mathbf r,t).
$$
This sees a lot of use in approximate methods in quantum mechanics, primarily in the study of Bose-Einstein condensates. However, it's also very useful in describing nonlinear phenomena in waves, such as e.g. intense laser beams in fibers, water waves in canals, and so on.
A: The best uses of linear theories is to describe simple systems in which the ratio of input:output is constant.
Eg if the input doubles, the output doubles.
Linear systems are time independent in the sense that the state of the system before doesn't impact on the state of the system now and in future. Putting this less abstractly: Linear systems don't have positive or negative feedback. The outputs of a linear system do not feedback into the inputs of a
the same linear system.
Linear systems work well for describing simple systems. As the quantum realm is the very small, linear equations seem to describe the simple interactions.
General relativity is non-linear. This means that it's time dependent and the ratio of input:output doesn't remain constant.
Many phenomena in nature require nonlinear description including: the spread of viruses (covid), the weather, etc. Here, the processes going on are emergent, often contain elements of chaos, fractal geometry, and self-organisation.
It will be necessary to explain how simple quantum phenomena combine to generate more complex systems with emergent properties. If that doesn't happen, we will not be able to understand how the small scale connects to large macro scale events such as those described in GR.
Therefore, a non-linear formulation of quantum mechanics is probably one of the logical steps forward in the pursuit of a connecting framework between QM and GR. Whether it turns out to be the approach that works, time will tell.
One of the issues is that much of physics to date have been studies of simple systems.  This is logical because until you understand the simple processes, understanding complex phenomenon is a stretch. However, now more and more Physics researchers are turning to non-linear mathematics to study nature. We have reached an impasse in QM and need to think outside the box to find s solution.
Tim Palmer of Oxford University has made early attempts to do this, but Tim's day job is to understand weather, not to study quantum mechanics. However,  it's become one of his side-interests. Others are now engaging with nonlinear QM. Hopefully,  21st century Physics will be primarily a non-linear dominated field, thus allowing humanity to understand the more complex emergent systems that dominate our environment.
Chaos theory and QM
QM non-computable not random
