We present the first search for an electrically charged resonance $W^\prime$ decaying to a WZ boson pair using 4.1 fb$^{-1}$ of integrated luminosity collected with the D0 detector at the Fermilab Tevatron $p\bar{p}$ collider. The WZ pairs are reconstructed through their decays into three charged leptons ($\ell=e,\mu$). A total of 9 data events is observed in good agreement with the background prediction. We set 95% C.L. limits on the W'WZ coupling and on the W' production cross section multiplied by the branching fractions. We also exclude W' masses between 188 and 520 GeV within a simple extension of the standard model and set the most restrictive limits to date on low-scale technicolor models.