Yes, kinda, the core which contributes (estimated at about a third) to the overall mass of the Earth
The Earth exerts a gravitational force on everything in the universe equivalent to:
Fgrav = G * M_Earth * m_object / (distance from center of Earth to center of object)^2
So, the Earth tries to pull things toward it, but typically the distance is so great that the object barely notices.
Also note, that this means everything in the universe is trying to pull the Earth toward it.
So, Fgrav on Earth is typically assigned a value of g = 9.81 m/s^2 for convenience as the value will not vary much, unless the object is particularly massive or particularly distant from the surface of our planet.
Not an expert, but gravity is not an actual field.
It somewhat seems like one because, as you can see from the formula, gravity is inversely proportionate to distance squared.
In theory, this means gravity has no limit to its reach; it just becomes infinitesimally small/weak.
Magnetism is a true field which again in theory is infinite, but gets weak as the distance from "center" of field decreases.
As I recall, the field strength diminishes rapidly and basically becomes unmeasurable.
Anyone with more extensive knowledge, feel free to correct, expand, etc.