  We prove a general sparse domination theorem in a space of homogeneous type,
in which a vector-valued operator is controlled pointwise by a positive, local
expression called a sparse operator. We use the structure of the operator to
get sparse domination in which the usual $\ell^1$-sum in the sparse operator is
replaced by an $\ell^r$-sum. This sparse domination theorem is applicable to
various operators from both harmonic analysis and (S)PDE. Using our main
theorem, we prove the $A_2$-theorem for vector-valued Calder\'on--Zygmund
operators in a space of homogeneous type, from which we deduce an anisotropic,
mixed norm Mihlin multiplier theorem. Furthermore, we show quantitative
weighted norm inequalities for the Rademacher maximal operator, for which
Banach space geometry plays a major role.
