  We report results from a deep polarization imaging of the nearby radio galaxy
3C$\,$84 (NGC$\,$1275). The source was observed with the Global Millimeter VLBI
Array (GMVA) at 86$\,$GHz at an ultra-high angular resolution of $50\mu$as
(corresponding to 250$R_{s}$). We also add complementary multi-wavelength data
from the Very Long Baseline Array (VLBA; 15 & 43$\,$GHz) and from the Atacama
Large Millimeter/submillimeter Array (ALMA; 97.5, 233.0, and 343.5$\,$GHz). At
86$\,$GHz, we measure a fractional linear polarization of $\sim2$% in the VLBI
core region. The polarization morphology suggests that the emission is
associated with an underlying limb-brightened jet. The fractional linear
polarization is lower at 43 and 15$\,$GHz ($\sim0.3-0.7$% and $<0.1$%,
respectively). This suggests an increasing linear polarization degree towards
shorter wavelengths on VLBI scales. We also obtain a large rotation measure
(RM) of $\sim10^{5-6}~{\rm rad/m^{2}}$ in the core at $\gtrsim$43$\,$GHz.
Moreover, the VLBA 43$\,$GHz observations show a variable RM in the VLBI core
region during a small flare in 2015. Faraday depolarization and Faraday
conversion in an inhomogeneous and mildly relativistic plasma could explain the
observed linear polarization characteristics and the previously measured
frequency dependence of the circular polarization. Our Faraday depolarization
modeling suggests that the RM most likely originates from an external screen
with a highly uniform RM distribution. To explain the large RM value, the
uniform RM distribution, and the RM variability, we suggest that the Faraday
rotation is caused by a boundary layer in a transversely stratified jet. Based
on the RM and the synchrotron spectrum of the core, we provide an estimate for
the magnetic field strength and the electron density of the jet plasma.
