  We explore the structure of the derived spherical Hecke algebra of a $p$-adic
group, a graded associative algebra whose degree $0$ subalgebra is the
classical spherical Hecke algebra. Working with $\mathbb Z/ p^a$ coefficients,
we establish a Satake homomorphism relating this graded algebra to the
corresponding graded algebra for the torus. We investigate the image of this
homomorphism in degree $1$, as well as other properties, such as transitivity
with respect to inclusion of Levi subgroups.
