The dynamical properties of water molecules confined in the unidirectional hydrophilic nanopores of AlPO4-54 are investigated with quasi-elastic neutron scattering as a function of temperature down to 118 K. AlPO4-54 has among the largest pores known for aluminophosphates and zeolites (about 1.3 nm), though they are small enough to prevent water crystallization due to the high degree of confinement. Water molecular diffusion into the pore is here measured down to 258 K. Diffusion is slower than in bulk water and has an activation energy of Ea = (20.8 ± 2.8) kJ/mol, in agreement with previous studies on similar confining media. Surprisingly, local hydrogen dynamics associated with water reorientation is measured down to temperatures (118 K), i.e., well below the expected glass transition temperature of bulk water. The reorientational time scale shows the well-known non-Arrhenius behavior down to the freezing of water mass diffusion, while it shows a feeble temperature dependence below. This fast local dynamics, of the order of fractions of nanoseconds, is believed to take place in the dense, highly disordered amorphous water occupying the pore center, indicating its possible plastic nature.