On a compact Riemannian manifold with boundary, we prove a spectral inequality for the bi-Laplace operator with so-called ``clamped'' boundary conditions, that is, homogeneous Dirichlet and Neumann conditions simultaneously. This spectral inequality allows one to observe finite sums of eigenfunctions for this fourth-order elliptic operator, from an arbitrary open subset of the manifold. Moreover, the constant that appears in the inequality grows as $\exp( C \mu^{1/4})$ where $\mu$ is the largest eigenvalue associated with the eigenfunctions appearing in the sum. This type of inequality is known for the Laplace operator. As an application, we obtain a null-controllability result for a higher-order parabolic equation. The proof is based on the derivation of different types of Carleman estimates for an elliptic operator related to the bi-Laplace operator: in the interior and at some boundaries. One of these estimates exhibits a loss of one full derivative. Its proof requires the introduction of an appropriate semi-classical calculus and a delicate microlocal argument.