A search for a light charged Higgs boson (${H}^{+}$) decaying to a $W$ boson and a $CP$-odd Higgs boson ($A$) in final states with $e\ensuremath{\mu}\ensuremath{\mu}$ or $\ensuremath{\mu}\ensuremath{\mu}\ensuremath{\mu}$ is performed using data from $pp$ collisions at $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$, recorded by the CMS detector at the LHC and corresponding to an integrated luminosity of $35.9\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. In this search, it is assumed that the ${H}^{+}$ boson is produced in decays of top quarks, and the $A$ boson decays to two oppositely charged muons. The presence of signals for ${H}^{+}$ boson masses between 100 and 160 GeV and $A$ boson masses between 15 and 75 GeV is investigated. No evidence for the production of the ${H}^{+}$ boson is found. Upper limits at 95% confidence level are obtained on the combined branching fraction for the decay chain, $t\ensuremath{\rightarrow}b{H}^{+}\ensuremath{\rightarrow}b{W}^{+}A\ensuremath{\rightarrow}b{W}^{+}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$, of $1.9\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ to $8.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, depending on the masses of the ${H}^{+}$ and $A$ bosons. These are the first limits for these decay modes of the ${H}^{+}$ and $A$ bosons.