We report on the long-range antiferromagnetic ordering with two successive phase transitions at ${T}_{\mathrm{N}1}=4.0\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, ${T}_{\mathrm{N}2}\phantom{\rule{4pt}{0ex}}\ensuremath{\approx}1.3\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ and the heavy-fermion-like behaviors in PrPdAl, a Kondo lattice with kagome-like structure of non-Kramers Pr ions in the hexagonal basal plane. The ground-state antiferromagnetism forms by two-thirds Pr ions with nearly full magnetic moment and one-third with strongly frustrated and reduced moment. The easy magnetization axis points along $c$ whereas the ordered Pr $4f$ moments reside in the basal plane, mirroring the important effect of spin frustrations. The electronic specific-heat coefficient is strongly enhanced to $\ensuremath{\sim}940\phantom{\rule{0.16em}{0ex}}\mathrm{mJ}/\mathrm{mol}\phantom{\rule{4pt}{0ex}}{\mathrm{K}}^{2}$, and the Kadowaki-Woods ratio is factor-of-25 smaller than that of conventional heavy-fermion compounds with Kramers doublet ground state. The effective Gr\"uneisen ratio estimated from thermal expansion reveals large values in the vicinity of the broadened phase transition at ${T}_{\mathrm{N}2}$, manifesting its instability against pressure. Accordingly, PrNiAl, a chemically pressurized analog of the titled compound, was found to reveal only one antiferromagnetic transition at a considerably higher temperature of 6.9 K. These results demonstrate that PrPdAl and related homologues can offer excellent cases for investigating competing ground states in frustrated Kondo lattices formed by non-Kramers magnetic ions.