We present a Bayesian hierarchical modelling approach to infer the cosmic matter density field, and the lensing and the matter power spectra, from cosmic shear data. This method uses a physical model of cosmic structure formation to infer physically plausible cosmic structures, which accounts for the non-Gaussian features of the gravitationally evolved matter distribution and light-cone effects. We test and validate our framework with realistic simulated shear data, demonstrating that the method recovers the unbiased matter distribution and the correct lensing and matter power spectrum. While the cosmology is fixed in this test, and the method employs a prior power spectrum, we demonstrate that the lensing results are sensitive to the true power spectrum when this differs from the prior. In this case, the density field samples are generated with a power spectrum that deviates from the prior, and the method recovers the true lensing power spectrum. The method also recovers the matter power spectrum across the sky, but as currently implemented, it cannot determine the radial power since isotropy is not imposed. In summary, our method provides physically plausible inference of the dark matter distribution from cosmic shear data, allowing us to extract information beyond the two-point statistics and exploiting the full information content of the cosmological fields.