In this paper, we prove the Novikov conjecture for a class of highly non-linear groups, namely discrete subgroups of the diffeomorphism group of a compact smooth manifold. This removes the volume-preserving condition in a previous work. This result is proved by studying operator K-theory and group actions on continuous fields of infinite dimensional non-positively curved spaces.Proof. It follows from [AAS20, Section 5] that the above homomorphism gives rise to an isomorphism between KK Γ R, * (EΓ, A) and KK Γ R, * (EΓ, A) τ , which is a subgroup of KK Γ R, * (EΓ, A) called its τ -part. Lemma 2.9. Let Γ be a discrete group, let X be a free and proper Γ-space, let A and B be two Γ-C * -algebras, and let ϕ : A → B be a *homomorphism that is Γ-equivariant and a homotopy equivalence, i.e., there exists a (possibly non-equivariant) * -homomorphism ψ : B → A such that ψ • ϕ and ϕ • ψ are homotopic to the identity maps on A and B, respectively. Then the homomorphismProof. This uses a cutting-and-pasting argument similar to [GWY21, Lemma 8.3].2.2. Hilbert-Hadamard spaces. In this section, we review some basics of Hilbert-Hadamard spaces (cf. [GWY21, Section 3]).Definition 2.10. A metric space (X, d) is CAT(0) if for any p, q, r, m ∈ X satisfying d(q, m) = d(r, m) = 1 2 d(q, r), the following CN inequality