Search for the charged lepton flavor violating decay <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="TIMEQ1"><mml:mi>ψ</mml:mi></mml:math></inline-formula>(3686)<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="TIMEQ2"><mml:mi> </mml:mi><mml:mstyle mathvariant="bold"><mml:mrow/><mml:msup><mml:mrow><mml:mi> </mml:mi><mml:mi>e</mml:mi></mml:mrow><mml:mi> </mml:mi></mml:msup><mml:mstyle mathvariant="bold"><mml:mo>±</mml:mo><mml:msup><mml:mrow><mml:mi> </mml:mi><mml:mi>μ</mml:mi></mml:mrow><mml:mi> </mml:mi></mml:msup><mml:mstyle mathvariant="bold"><mml:mo>∓</mml:mo></mml:mstyle></mml:mstyle></mml:mstyle></mml:math></inline-formula>

By analyzing $(2367.0\pm11.1)\times10^6$ $\psi(3686)$ events collected in $e^+e^-$ collisions at $\sqrt{s}=3.686~\rm~GeV$ with the BESIII detector at the BEPCII collider, we report the first search for the charged lepton flavor violating decay $\psi(3686)\to~e^{\pm}\mu^{\mp}$. No signal is observed. An upper limit on the branching fraction $\mathcal{B}(\psi(3686)\to~e^{\pm}\mu^{\mp})$ is determined to be $1.4\times10^{-8}$ at the 90% confidence level.