  We report a site-selective ^{17}O spin-lattice relaxation rate T_1^{-1} in
the vortex state of underdoped YBa_2Cu_4O_8. We found that T_1^{-1} at the
planar sites exhibits an unusual nonmonotonic NMR frequency dependence. In the
region well outside the vortex core, T_1^{-1} cannot be simply explained by the
density of states of the Doppler-shifted quasiparticles in the d-wave
superconductor. Based on T_1^{-1} in the vortex core region, we establish
strong evidence that the local density of states within the vortex core is
strongly reduced.
