  We show that Hubbard models with nearest-neighbor hopping and a
nearest-neighbor hardcore constraint exhibit `maximal' Hilbert space
fragmentation in many lattices of arbitrary dimension $d$. Focusing on the
$d=1$ rhombus chain and the $d=2$ Lieb lattice, we demonstrate that the
fragmentation is strong for all fillings in the thermodynamic limit, and
explicitly construct all emergent integrals of motion, which include an
extensive set of higher-form symmetries. Blockades consisting of frozen
particles partition the system in real space, leading to anomalous dynamics.
Our results are potentially relevant to optical lattices of dipolar and
Rydberg-dressed atoms.
