  The purpose of this paper is to study property (RD) for locally compact Hecke
pairs. We discuss length functions on Hecke pairs and the growth of Hecke
pairs. We establish an equivalence between property (RD) of locally compact
groups and property (RD) of certain locally compact Hecke pairs. This allows us
to transfer several important results concerning property (RD) of locally
compact groups into our setting, and consequently to identify many classes of
examples of locally compact Hecke pairs with property (RD). We also show that a
reduced discrete Hecke pair $(G,H)$ has (RD) if and only if its Schlichting
completion $\bar{G}$ has (RD). Then it follows that the relative unimodularity
is a necessary condition for a discrete Hecke pair to possess property (RD).
