For the action of a locally compact and totally disconnected group G (the most important examples of such being all discrete groups as well as all p-adic reductive groups) on a pair of locally compact spaces X and Y we construct, by sheaf theoretic means, a new equivariant and bivariant cohomology theory. In contrast to the classical Borel construction of equivariant cohomology our construction is not localized in the unit element of the group. If we take for the first space Y "the" universal proper G-action then we obtain for the second space its delocalized equivariant homology. All this is in exact formal analogy to the definition of equivariant K-homology by Baum, Connes, Higson starting from the bivariant equivariant Kasparov KK-theory. Under certain basic finiteness conditions on the first space Y we conjecture the existence of a Chern character from the equivariant Kasparov KK-theory of Y and X into our cohomology theory made two-periodic which becomes an isomorphism upon tensoring the KK-theory with the complex numbers. This conjecture is proved in this paper for profinite groups G. An essential role in our construction is played by a bivariant version of Segal localization which we establish for KK-theory.