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We report results from a deep polarization imaging of the nearby radio galaxy3C$\,$84 (NGC$\,$1275). The source was observed with the Global Millimeter VLBIArray (GMVA) at 86$\,$GHz at an ultra-high angular resolution of $50\mu$as(corresponding to 250$R_{s}$). We also add complementary multi-wavelength datafrom the Very Long Baseline Array (VLBA; 15 & 43$\,$GHz) and from the AtacamaLarge Millimeter/submillimeter Array (ALMA; 97.5, 233.0, and 343.5$\,$GHz). At86$\,$GHz, we measure a fractional linear polarization of $\sim2$% in the VLBIcore region. The polarization morphology suggests that the emission isassociated with an underlying limb-brightened jet. The fractional linearpolarization is lower at 43 and 15$\,$GHz ($\sim0.3-0.7$% and $<0.1$%,respectively). This suggests an increasing linear polarization degree towardsshorter 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 coreregion during a small flare in 2015. Faraday depolarization and Faradayconversion in an inhomogeneous and mildly relativistic plasma could explain theobserved linear polarization characteristics and the previously measuredfrequency dependence of the circular polarization. Our Faraday depolarizationmodeling suggests that the RM most likely originates from an external screenwith a highly uniform RM distribution. To explain the large RM value, theuniform RM distribution, and the RM variability, we suggest that the Faradayrotation is caused by a boundary layer in a transversely stratified jet. Basedon the RM and the synchrotron spectrum of the core, we provide an estimate forthe magnetic field strength and the electron density of the jet plasma.

Spatially resolved origin of mm-wave linear polarization in the nuclear region of 3C 84