Experimental determination of the neutron resonance peak of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>Er</mml:mi><mml:mprescripts/><mml:none/><mml:mn>162</mml:mn></mml:mmultiscripts></mml:math> at 67.8 eV

$^{162}\mathrm{Er}$ is one of the $p$ nuclei in nuclear astrophysics, and its ($n,\ensuremath{\gamma}$) cross sections are important input parameters in nuclear astrophysics network calculations. Resonance data for $^{162}\mathrm{Er}(n,\ensuremath{\gamma})^{163}\mathrm{Er}$ at 67.8 eV is currently not present in the EXFOR database; however, it is included in the ENDF/B VIII.0 database. The ($n,\ensuremath{\gamma}$) cross section of $^{\mathrm{nat}}\mathrm{Er}$ has been measured in the energy range of 1--100 eV at the Back-n facility in the China Spallation Neutron Source. However, due to the influence of the in-beam $\ensuremath{\gamma}$ background of the Back-n Facility, it is has not been possible to observe the neutron resonance peak of $^{162}\mathrm{Er}$ at 67.8 eV. A general simulation method is proposed for quantifying the in-beam $\ensuremath{\gamma}$-ray background. By subtracting the in-beam $\ensuremath{\gamma}$-ray background, the neutron-capture yields of $^{162}\mathrm{Er}$ are obtained within the energy range of 20--100 eV. The neutron resonance parameters of $^{162}\mathrm{Er}$ at 67.8 eV are then extracted successfully. It is found that ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\gamma}}=101.15\ifmmode\pm\else\textpm\fi{}10.08$ meV and ${\mathrm{\ensuremath{\Gamma}}}_{n}=2.79\ifmmode\pm\else\textpm\fi{}0.28$ meV, which are consistent with the ENDF/B VIII.0 database. However, the 51.4-eV resonance peak of $^{162}\mathrm{Er}(n,\ensuremath{\gamma}$) is currently unobserved.