Using a variational procedure within the effective-mass approximation we calculate the binding energy of a shallow donor impurity in cylindrical GaAs low-dimensional systems, under the action of an electric field applied in the axial direction, and considering an infinite confinement potential. We calculate the dependence of the binding energy on the system geometry, the applied electric field, and the donor-impurity position. Our results for small fields and large radius as well as for length greater than the radius coincide quite well with previous results in quantum wells, quantum-well wires, and quantum dots. We find that a proper knowledge of the impurity distribution inside the GaAs low-dimensional system is of relevance in a qualitative comparison between theoretical and experimental results concerning the binding energy of shallow impurities under the action of an applied electric field.