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We report on a study of the process ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}{(D{\overline{D}}^{*})}^{\ensuremath{\mp}}$ at $\sqrt{s}=4.26\text{ }\text{ }\mathrm{GeV}$ using a $525\text{ }\mathrm{pb}{}^{\ensuremath{-}1}$ data sample collected with the BESIII detector at the BEPCII storage ring. A distinct charged structure is observed in the ${(D{\overline{D}}^{*})}^{\ensuremath{\mp}}$ invariant mass distribution. When fitted to a mass-dependent-width Breit-Wigner line shape, the pole mass and width are determined to be ${M}_{\mathrm{pole}}=\phantom{\rule{0ex}{0ex}}\mathbf{(}3883.9\ifmmode\pm\else\textpm\fi{}1.5\phantom{\rule{0.333em}{0ex}}(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}4.2\phantom{\rule{0.333em}{0ex}}(\mathrm{syst})\mathbf{)}\text{ }\text{ }\mathrm{MeV}/{c}^{2}$ and ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{pole}}=\mathbf{(}24.8\ifmmode\pm\else\textpm\fi{}3.3\phantom{\rule{0.333em}{0ex}}(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}11.0\phantom{\rule{0.333em}{0ex}}(\mathrm{syst})\mathbf{)}\text{ }\text{ }\mathrm{MeV}$. The mass and width of the structure, which we refer to as ${Z}_{c}(3885)$, are $2\ensuremath{\sigma}$ and $1\ensuremath{\sigma}$, respectively, below those of the ${Z}_{c}(3900)\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}J/\ensuremath{\psi}$ peak observed by BESIII and Belle in ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}J/\ensuremath{\psi}$ final states produced at the same center-of-mass energy. The angular distribution of the $\ensuremath{\pi}{Z}_{c}(3885)$ system favors a ${J}^{P}={1}^{+}$ quantum number assignment for the structure and disfavors ${1}^{\ensuremath{-}}$ or ${0}^{\ensuremath{-}}$. The Born cross section times the $D{\overline{D}}^{*}$ branching fraction of the ${Z}_{c}(3885)$ is measured to be ${\ensuremath{\sigma}\mathbf{(}{e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}\phantom{\rule{0ex}{0ex}}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}{Z}_{c}(3885)}^{\ensuremath{\mp}}\mathbf{)}\ifmmode\times\else\texttimes\fi{}{\mathcal{B}\mathbf{(}{Z}_{c}(3885)}^{\ensuremath{\mp}}\ensuremath{\rightarrow}{(D{\overline{D}}^{*})}^{\ensuremath{\mp}}\mathbf{)}=\mathbf{(}83.5\ifmmode\pm\else\textpm\fi{}6.6\phantom{\rule{0.333em}{0ex}}(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}22.0\phantom{\rule{0.333em}{0ex}}(\mathrm{syst})\mathbf{)}\text{ }\text{ }\text{ }\mathrm{pb}$. Assuming the ${Z}_{c}(3885)\ensuremath{\rightarrow}D{\overline{D}}^{*}$ signal reported here and the ${Z}_{c}(3900)\ensuremath{\rightarrow}\ensuremath{\pi}J/\ensuremath{\psi}$ signal are from the same source, the partial width ratio $(\mathrm{\ensuremath{\Gamma}}({Z}_{c}(3885)\ensuremath{\rightarrow}D{\overline{D}}^{*})/\mathrm{\ensuremath{\Gamma}}({Z}_{c}(3900)\ensuremath{\rightarrow}\ensuremath{\pi}J/\ensuremath{\psi}))=6.2\ifmmode\pm\else\textpm\fi{}1.1\phantom{\rule{0.333em}{0ex}}(\mathrm{stat})\phantom{\rule{0.333em}{0ex}}\ifmmode\pm\else\textpm\fi{}2.7\phantom{\rule{0.333em}{0ex}}(\mathrm{syst})$ is determined.

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