We present a search for $Wb\overline{b}$ production in $p\overline{p}$ collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{T}\mathrm{e}\mathrm{V}$ in events containing one electron, an imbalance in transverse momentum, and two $b$-tagged jets. Using $174\text{ }\text{ }{\mathrm{p}\mathrm{b}}^{\ensuremath{-}1}$ of integrated luminosity accumulated by the D0 experiment at the Fermilab Tevatron collider, and the standard-model description of such events, we set a 95% C.L. upper limit on $Wb\overline{b}$ production of 6.6 pb for $b$ quarks with transverse momenta ${p}_{T}^{b}>20\text{ }\text{ }\mathrm{G}\mathrm{e}\mathrm{V}$ and $b\overline{b}$ separation in pseudorapidity-azimuth space $\ensuremath{\Delta}{\mathcal{R}}_{bb}>0.75$. Restricting the search to optimized $b\overline{b}$ mass intervals provides upper limits on $WH$ production of 9.0--12.2 pb for Higgs-boson masses of 105--135 GeV.