  In heavy-fermion systems with $f$ electrons, there is an intricate interplay
between Kondo screening and magnetic correlations, which can give rise to
various exotic phases. Recently, similar interplay appears to also occur in
$d$-electron systems, but the underlying mechanism remains elusive. Here, using
inelastic neutron scattering, we investigate the temperature evolution of the
low-energy spin waves in a metallic van der Waals ferromagnet
Fe$_{3-x}$GeTe$_{2}$ (Curie temperature $T_{\rm C}\sim160$ K), where the
Kondo-lattice behavior emerges in the ferromagnetic phase below a
characteristic temperature $T^*\sim90$ K. We observe that the magnon damping
constant diverges at both low and high temperatures, exhibiting a minimum
coincidentally around $T^*$. Such an observation is analogous to the
resistivity minimum as due to the single-impurity Kondo effect. This unusual
behavior is described by a formula that combines logarithmic and power terms,
representing the dominant contributions from Kondo screening and thermal
fluctuations, respectively. Furthermore, we find that the magnon damping
increases with momentum below $T_{\rm C}$. These findings can be explained by
considering spin-flip electron-magnon scattering, which serves as a magnonic
analog of the Kondo-impurity scattering, and thus provides a measure of the
Kondo coupling through magnons. Our results provide critical insights into how
Kondo coupling manifests itself in a system with magnetic ordering and shed
light on the coexistence of and interplay between magnetic order and Kondo
effect in itinerant 3$d$-electron systems.
