Q:

Do physical fields always have dimension of the Tangent Space at a point?

Say we only consider classical fields in 3 dimensions. In a 3 dimensional space you have scalar or vector fields, where scalar fields can be understood as vectors of one dimension. The other physical vector Fields I know of have three dimensions. What about the intermediate? Can 2 dimensional vector fields exist in 3D space and have physical meaning?

A:

The spin of a particle is an example of a kind of a 2-dimensional vector field in 3D. 
Any measurement results in a 1/2 or -1/2 value, and the particle has after it, a state vector "up" or "down". 
I said "a kind" of vector, because the components of a 3D vector are modified after a coordinate rotation following rules that are different for the spin state vectors. 
That is why they are called spinors instead of vectors.  

