  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.
