The recently synthesized ThFeAsN iron-pnictide superconductor exhibits a $T_c$ of 30 K, the highest of the 1111-type series in absence of chemical doping. To understand how pressure affects its electronic properties, we carried out microscopic investigations up to 3 GPa via magnetization, nuclear magnetic resonance, and muon-spin rotation experiments. The temperature dependence of the ${}^{75}$As Knight shift, the spin-lattice relaxation rates, and the magnetic penetration depth suggest a multi-band $s^{pm}$-wave gap symmetry in the dirty limit, while the gap-to-$T_c$ ratio $Delta/k_mathrm{B}T_c$ hints at a strong-coupling scenario. Pressure modulates the geometrical parameters, thus reducing $T_c$, as well as $T_m$, the temperature where magnetic-relaxation rates are maximized, both at the same rate of approximately -1.1 K/GPa. This decrease of $T_c$ with pressure is consistent with band-structure calculations, which relate it to the deformation of the Fe 3$d_{z^2}$ orbitals.
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