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The nuclear level density is a fundamental quantity in nuclear physics, governing various nuclear reactions and astrophysical processes. In this study, we report on the collective enhancement of nuclear level density and its fade-out with excitation energy in the deformed $^{161}$Dy, obtained through an exclusive measurement of neutron evaporation spectra. The $^{162}$Dy nucleus was populated via the transfer of a triton in a $^{7}$Li-induced reaction on $^{159}$Tb. Statistical model analysis of the neutron spectra revealed a large collective enhancement factor of 42$\pm$2, consistent with microscopic calculations. This enhancement factor is similar to the one obtained for mass A$\sim$170 in our previous measurement. The energy-dependent collective enhancement over a wide range of excitation energies was inferred by combining the present results with the available Oslo level density below the neutron binding energy.