Molecular dynamics simulations for a TIP4P water cluster consisting of 32 molecules at T=200 K, under the influence of a broad range of constant electric fields (0.5-7.0x10(7) V/cm), are presented. This work focuses on the evolution of the single particle translational dynamics, mainly along the field axis as the field is progressively increased, by means of mean-square-displacement curves, the self-part of the van Hove distribution functions and the intermediate scattering functions. Two critical fields have been identified, the one, (E-C1=1.5x10(7) V/cm) assigned to the onset of the dipole alignment and the second one (E-C2=5.0x10(7) V/cm) to the onset of crystallization. These transitions are marked by an abrupt increase of the corresponding structure relaxation times, which remain nearly constant for electric fields between E-C1 and E-C2. Structure relaxation has been found to obey stretched exponential dynamics, whereas the Q dependence of the relaxation times, for all fields, followed a power law. Fields weaker than E-C1 have been found to induce a weakening of the molecular interactions. In this case, the system develops a dynamic behavior similar to that met in the liquid.