  Based on specific heat and magnetoresistance measurements, we report that a
"heavy" electronic state exists below $T \approx$ 20 K in KNi$_2$Se$_2$, with
an increased carrier mobility and enhanced effective electronic band mass, $m$*
= 6$m_b$ to 18$m_b$. This "heavy" state evolves into superconductivity at $T_c$
= 0.80(1) K. These properties resemble that of a many-body heavy-fermion state,
which derives from the hybridization between localized magnetic states and
conduction electrons. Yet, no evidence for localized magnetism or magnetic
order is found in KNi$_2$Se$_2$ from magnetization measurements or neutron
diffraction. Instead, neutron pair-distribution-function analysis reveals the
presence of local charge-density-wave distortions that disappear on cooling, an
effect opposite to what is typically observed, suggesting that the
low-temperature electronic state of KNi$_2$Se$_2$ arises from cooperative
Coulomb interactions and proximity to, but avoidance of, charge order.
