  We offer a systematic study of rigid analytic motives over general rigid
analytic spaces, and we develop their six-functor formalism. A key ingredient
is an extended proper base change theorem that we are able to justify by
reducing to the case of algebraic motives. In fact, more generally, we develop
a powerful technique for reducing questions about rigid analytic motives to
questions about algebraic motives, which is likely to be useful in other
contexts as well. We pay special attention to establishing our results without
noetherianity assumptions on rigid analytic spaces. This is indeed possible
using Raynaud's approach to rigid analytic geometry.
