We present a search for the decay $X(3872) \to \pi^+\pi^-\pi^0$ in the $(772\pm11)\times10^6$ $\Upsilon(4S)\to B \bar B$ data sample collected at the Belle detector, where the $X(3872)$ is produced in $B^{\pm}\to K^{\pm}X(3872)$ and $B^{0}\to K_{S}^0 X(3872)$ decays. We do not observe a signal, and set 90% credible upper limits for two different models of the decay processes: if the decay products are distributed uniformly in phase space, $\mathcal{B}(X(3872) \to \pi^+\pi^-\pi^0) < 1.3\%$; if $M(\pi^+\pi^-)$ is concentrated near the mass of the $D^0 \bar D^0$ pair in the process $X(3872)\to D^0\bar{D}^{*0}+c.c.\to D^0 \bar D^{0}\pi^0\to\pi^+ \pi^- \pi^0$, $\mathcal{B}(X(3872) \to \pi^+\pi^-\pi^0) < 1.2\times10^{-3}$.
