  We present a new high-precision parametric strong lensing model of the galaxy
cluster MACS J0416.1-2403, at z=0.396, which takes advantage of the MUSE Deep
Lensed Field (MDLF), with 17.1h integration in the northeast region of the
cluster, and Hubble Frontier Fields data. We spectroscopically identify 182
multiple images from 48 background sources at 0.9<z<6.2, and 171 cluster member
galaxies. Several multiple images are associated to individual clumps in
multiply lensed resolved sources. By defining a new metric, which is sensitive
to the gradients of the deflection field, we show that we can accurately
reproduce the positions of these star-forming knots despite their vicinity to
the model critical lines. The high signal-to-noise ratio of the MDLF spectra
enables the measurement of the internal velocity dispersion of 64 cluster
galaxies, down to m(F160W)=22. This allowed us to independently estimate the
contribution of the subhalo mass component of the lens model from the measured
Faber-Jackson scaling relation. Our best reference model, which represents a
significant step forward compared to our previous analyses, was selected from a
comparative study of different mass parametrizations. The root-mean-square
displacement between the observed and model-predicted image positions is only
0.40", which is 33% smaller than in all previous models. The mass model appears
to be particularly well constrained in the MDLF region. We characterize the
robustness of the magnification map at varying distances from the model
critical lines and the total projected mass profile of the cluster.

Astrophysics of Galaxies;Cosmology and Nongalactic Astrophysics