Is the Lorentz force a vector field or just a vector? I've heard both yes and no. 
Is the Lorentz force a vector field or just a vector?
$$\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$$
 A: @rushinc1 nearly had it.
The Lorentz force is sadly not a vector field in the normal sense (i.e. a smooth mapping from $(\vec r: \mathbb R^3, t : \mathbb R) \to \mathbb R^3$), as you can see from the explicit presence of $\vec v$ in its definition: you need to know the velocity of the particle in order to calculate the magnetic force upon it, and that information is not contained merely in this information about where and when the Lorentz force is being evaluated.
It is sadly also not a 2-tensor field in the normal sense as its transformation of a velocity vector is not linear in that velocity vector (it is a sort of affine transformation because the $\vec E$ part is a sort of fixed constant). 
However in special relativity it does become an antisymmetric 2-tensor field on the space of 4-vectors; the presence of the extra time component in the velocity 4-vector gives the tensor a perfect place to inject the $\vec E$ field alongside the $\vec B$ field. You get the antisymmetric 2-tensor $\partial_\alpha A_\beta - \partial_\beta A_\alpha$ (where $A^\mu$ is the standard 4-vector-potential) as a perfectly linear transformation of a four-velocity $U^\mu$ to a four-force $dp^\nu/d\tau,$ making it a clear 2-tensor.
Then in general relativity this again gets a little more complicated as there are many connections $\partial_\alpha$ to choose from, when we apply them to vectors, but the point is that in some generic sense anything that can be made out of well-defined 4-operators can be a 4-tensor. 
A: Mathematically a vector field is a vector valued function of spacetime coordinates only.  This makes it so we can talk about the source of a force independently from the object it acts on. 
In order to describe the Lorentz force on an individual (ie, "test") particle you need to know both its position AND its velocity (as well as its charge, of course).  You cannot divide the force by a unit ("test") velocity, because velocity is itself a vector.  At best I suppose one could establish a Lorentz tensor field to serve a similar purpose, but to try to describe the Lorentz force via a vector field is nonsensical.
Now, that being said, if you have a collection of particles (like a fluid) that fill all of space and you can assign a velocity vector to each point in space at each time then you can generate something like a Lorentz vector field.  However, this is really more of an engineering trick to solve fluid dynamics problems and I personally would not consider it a vector field in the same sense as gravity or the electromagnetic field.
A: In classical electromagnetism, the Lorentz force is just a vector and NOT a vector field. It can however be defined as a tensor field and that is exactly what it used when dealing with relativistic electromagnetism. 
A vector field is a function that assigns an $n$ dimensional vector to each point in an $n$ dimensional space. The Lorentz force cannot be defined by such a field as we need information about two $n$ dimensional vectors ($\mathbf{E}$ and $\mathbf{B}$) for a complete description of the force.
When relativistic effects are also considered, the Lorentz force can be defined uniquely by the equation
$$\frac{dp^{\alpha}}{d\tau} = qF^{\alpha\beta}U_{\beta}$$
Here $F^{\alpha\beta}$ is the electromagnetic tensor that can be considered equivalent to a "Lorentz field" of sorts.
