You can't meaningfully take the classical limit of the weak force, because at the scales where you could neglect the quantum interactions, the classical potential of the weak force is suppressed so strongly by the masses of the W- and Z- bosons that it is practically non-existent. This is because forces mediated by massive particles don't follow a pure inverse-square law in their classical limit, but have an additional factor $\exp(-\mu r)$ where $\mu$ is the mass of the mediator.
The weak force, like the strong force, is significant and meaningful only in its fully quantum description - there is no use in trying to describe classical aspects of it. But the notion of a "weak field" surrounding weak charges, just like the notion of a constant electric field surrounding an electric charge according to Coulomb's law, is a classical notion. There is a "weak field" in the sense that the quantum field whose quanta are the W- and Z-bosons is a potential fully analogous to the electromagnetic vector potential.
However, if you just formally take the classical limit and don't care for how physically meaningful it is, then the answer is that the "weak field" you would get would obey a distance law $\propto \frac{\exp(-\mu r)}{r^2}$ instead of the electromagnetic $\propto\frac{1}{r^2}$.
