Where does the electric field in light come from if it wasn't produced by a charge? If light is an electromagnetic radiation, then it has an electric field as one of its components. My question : where did that electric field come from if there was no charge around and electric fields can only be produced by charges. Also, if the electric field is produced by a charge, why it keeps propagating in space after the charge was long gone, why it doesn't vanish the moment the charge vanished?
My understanding of an electric field:
It's a mathematical quantity defined on every point of space that if you multiply it by a test charge will give you the amount of force on the test charge by the original charge which created the force.
I would prefer an explanation using my naive definition of the electric field without any additional assumptions.
 A: There are three sources to the electric field, and none is a changing magnetic field. The electric field at any point in space and time (w/o general relativistic considerations) is:
$$\vec E({\vec r, t})=\frac 1 {4\pi\epsilon_0}\int\Big[
\frac{\vec r-\vec r'}{|\vec r-\vec r'|^3}\rho(\vec r',t_r) +
\frac{\vec r-\vec r'}{|\vec r-\vec r'|^2}\frac 1 c \frac{\partial \rho(\vec r',t_r)}{\partial t} -
\frac{1}{|\vec r-\vec r'|}\frac 1 {c^2} \frac{\partial \vec J(\vec r',t_r)}{\partial t}
\Big]d^3{\vec r'}$$
where the retarded time:
$$t_r = t-\frac{|\vec r-\vec r'|} c $$
ensures the integration is over the past light-cone.
Inspection shows that the 3 sources of electric field are charge, changing charge, and changing current. That is it.
There is a similar expression for the magnetic field, showing that it is sourced by current and changing current.
You can take the solutions and show the $\vec E$ and $\vec B$ at source free points always satisfy the two acausal relations:
$$\vec{\nabla}\times E=-\frac{\partial\vec{B}}{\partial t}$$
$$\vec{\nabla}\times B=\frac 1{c^2}\frac{\partial\vec{E}}{\partial t}$$
The first one is commonly interpreted to mean changing magnetic field creates and electric field, but this incorrect (it would violate causality, for instance). What it really says is that a changing magnetic field is the negative of the curl of the electric field. Even if you are looking at the CMB , and the retarded time / $\vec r'$ in the source term refers to a proton / electron plasma 13.7 billion years ago/13.7 billion ly away (in some coordinate...that's why I said GR notwithstanding).
A: Your definition of electric field

It's a mathematical quantity defined on every point of space that if you multiply it by a test charge will give you the amount of force on the test charge by the original charge which created the force.

is too bound to the concept of electrostatic field to be useful when moving to electrodynamics.
Within electrostatics (or magnetostatics), the concept of field is just a proxy for the concept of force on charges due to other charges (or currents, in the case of magnetic fields). Being in a static condition, time does not enter the description. One can say that the presence of a charge at the position ${\bf r}$ is related to the presence of a force on a test charge at position ${\bf r'}$, or, in an equivalent way, that the force on the test charge is due to the presence of an electric field at the same position, due to the source charge.
Things become more complicated (and interesting) when there is some time variation.
Firstly, the force at time $t$ on a test charge at the position ${\bf r'}$ has to be related to the position of a source charge at a previous time $t'$. Secondly, not only the position of the source charge determines the force but also its movement and the time variation of possible currents. All these effects can be summarized in the Jefimenko's equations relating the fields to the charge and current sources (formula for the electric field is in JEB's answer).
At this point, the concept of field becomes more interesting than in the static case. Indeed, even if the field ${\bf E}({\bf r},t)$ depends on charge and currents at the previous time, if we know it at position ${\bf r}$ and time $t$, it is all we need for evaluating the force. Moreover, the wave-like character of the time-dependent fields allows us to determine the field evolution at each point if we know the fields at a previous time. This is important because we do not have to trace back the causal relations up to the original charges and currents originating the fields. Still, it is enough that we know enough about the fields immediately before the time we are interested.
The possibility of a causal chain of configurations is rooted in the wave-like time-dependent equations for the fields. Like in a mechanical wave, each configuration can be thought of as originating from the local evolution of the previous configuration. The same happens in the case of electromagnetic fields.
So, the question

why it keeps propagating in space after the charge was long gone, why it doesn't vanish the moment the charge vanished?

is equivalent to ask why, after throwing a stone in a pond, water waves keep propagating even after the stone is at rest at the bottom of the pond.  The answer is that water, in the case of the pond, and electromagnetic fields, in the case of charges, are dynamic systems by themselves. As such, they cease to be only mathematical concepts but acquire the status of physical systems. Such interpretation is completed by the fact that it is possible to associate properties like energy and momentum to the fields.
A: Electric (and magnetic) fields are always produced by the charges somewhere (not necessary locally). When you see light, you essentially see oscillating charges in the light source.
A: Light, or more generally EM radiation, consists of photons. There is no other way to produce EM radiation than from excited subatomic particles. All considerations of EM radiation as a perturbation of an overall existing EM field can be helpful for quantum mechanical processes around the atom (for which this mechanics was developed), but even then the perturbation happens from excited particles.
These emitters are particles with an electric field and a magnetic field. Please remember that both the electron and the other subatomic particles are intrinsic magnetic dipoles. They emit what they have; both electric field and magnetic field.
To me, a photon is the propagation of a portion of energy, with mutually induced and self-existing electric and magnetic field.

Where does the electric field in light come from if it wasn't produced by a charge?

Give an example of how light comes into the world without being emitted by subatomic particles. I cannot see such a possibility.

Also, if the electric field is produced by a charge, why it keeps propagating in space after the charge was long gone, why it doesn't vanish the moment the charge vanished?

The assumption that EM radiation is connected backwards to the source is not supported by experiments. Best example?: Consider that in annihilation processes the subatomic particles disappear completely. The photons emitted in these processes wobble away with their field components as long as they are not absorbed by other subatomic particles. They are independent of the source after they have left it.
