  The puckered lattice geometry, along with $p$-orbitals is often overlooked in
the study of topological physics. Here, we investigate the higher-order
topology of the $p_{x,y}$-orbital bands in acoustic metamaterials using a
simplified two-dimensional phosphorene lattice which possesses a puckered
structure. Notably, unlike the $s$-orbital bands in planar lattices, the unique
higher-order topology observed here is specific to $p$-orbitals and the
puckered geometry due to the unusual hopping patterns induced by them. {Using
acoustic pump-probe measurements in metamaterials}, we confirm the emergence of
the edge and corner states arising due to the unconventional higher-order
topology. We reveal the uniqueness of the higher-order topological physics here
via complimentary tight-binding calculations, finite-element simulations, and
acoustic experiments. We analyze the underlying physics of the special
properties of the edge and corner states in the puckered lattice acoustic
metamaterials from the picture of Wannier orbitals. Our work sheds light on the
intriguing physics of $p$-orbital topological physics in puckered lattices and
acoustic metamaterials which lead to unconventional topological boundary
states. \end{abstract}
