This paper treats finite Borel measures \(\mu\) on a separable \(C^ 3\) Banach manifold X and their logarithmic derivatives along vector fields. It is assumed that the Banach manifold X has a so-called Hilbert-Schmidt structure (a Hilbert space bundle \({\mathcal H}\) over X linearly and densely embedded in the tangent bundle TX such that any bounded linear map \(T_ xX\to {\mathcal H}_ x\) restricts to a Hilbert-Schmidt operator: \({\mathcal H}_ x\to {\mathcal H}_ x)\) and a connection compatible with it.
Then it is shown that each Borel measure \(\mu\) which is locally smooth along lines in the modelling Hilbert space H of \({\mathcal H}\) admits a logarithmic derivative along any vector field with values in \(TX^*\subset {\mathcal H}\subset TX\). Under further assumptions the \(L^ 2\)-norm of this logarithmic derivative with respect to \(\mu\) is given as an expression involving the Ricci curvature of the connection. Also the logarithmic derivative is well behaved under f-relatedness of vector fields and pullback of measures.