An enhanced low-energy electric dipole (E1) strength is identified for the weakly bound excited states of the neutron-rich isotope ^{27}Ne. The Doppler-shift lifetime measurements employing a combination of the γ-ray tracking array GRETINA, the plunger device, and the S800 spectrograph determine the lower limit of 0.030 e^{2} fm^{2} or 0.052 W.u. for the 1/2^{+}→3/2^{-} E1 transition in ^{27}Ne, representing one of the strongest E1 strengths observed among the bound discrete states in this mass region. This value is at least 30 times larger than that measured for the 3/2^{-} decay to the 3/2_{gs}^{+} ground state. A comparison of the present results to large-scale shell-model calculations points to an important role of core excitations and deformation in the observed E1 enhancement, suggesting a novel example of the electric dipole modes manifested in weakly bound deformed systems.