Hilbert field and hermitian Hilbert bundle

In the paper “Direct Images, Fields of Hilbert Spaces, and Geometric Quantization”, Lempert and Sz\H{o}ke proved that any flat analytic Hilbert field will induce a hermitian Hilbert bundle and gave an example of a flat Hilbert field that does not induce any Hilbert bundle. In this paper, we will provide an example of an analytic Hilbert field that does not induce any Hilbert bundle.