  Detecting and measuring a non-Gaussian signature of primordial origin in the
density field is a major science goal of next-generation galaxy surveys. The
signal will permit us to determine primordial physics processes and constrain
models of cosmic inflation. While traditional approaches utilise a limited set
of statistical summaries of the galaxy distribution to constrain primordial
non-Gaussianity, we present a field-level approach by Bayesian
forward-modelling the entire three-dimensional galaxy survey. Our method
naturally and fully self-consistently exploits the entirety of the large-scale
structure, e.g., higher-order statistics, peculiar velocity fields, and
scale-dependent galaxy bias, to extract information on the local
non-Gaussianity parameter, $\fnl$. We demonstrate the performance of our
approach through various tests with mock galaxy data emulating relevant
features of the \sdssiii{}-like survey, and additional tests with a
\textit{Stage IV} mock data set. These tests reveal that the method infers
unbiased values of $\fnl$ by accurately handling survey geometries, noise, and
unknown galaxy biases. We demonstrate that our method can achieve constraints
of $\sigma_{\fnl} \approx 8.78$ for \sdssiii{}-like data, an improvement of a
factor $\sim 2.5$ over currently published constraints. Tests with
next-generation mock data show that significant further improvements are
feasible with sufficiently high resolution. Furthermore, the results
demonstrate that our method can consistently marginalise all nuisance
parameters of the data model. The method further provides an inference of the
three-dimensional primordial density field, providing opportunities to explore
additional signatures of primordial physics.
