  In the dense metal-organic framework Na[Mn(HCOO)$_3$], Mn$^{2+}$ ions
($S=\frac{5}{2}$) occupy the nodes of a `trillium' hyperkagome net. We show
that this material exhibits a variety of behaviour characteristic of geometric
frustration: the N\'eel transition is suppressed well below the characteristic
magnetic interaction strength; short-range magnetic order persists far above
the N\'eel temperature; and the magnetic susceptibility exhibits a
pseudo-plateau at $\frac{1}{3}$-saturation magnetisation. We demonstrate that a
simple nearest-neighbour Heisenberg antiferromagnet model accounts
quantitatively for each observation, and hence Na[Mn(HCOO)$_3$] is the first
experimental realisation of this model on the trillium net. We develop a
mapping between this trillium model and that on the two-dimensional
Shastry-Sutherland lattice, and demonstrate how both link geometric frustration
within the classical spin liquid regime to a strong magnetocaloric response at
low fields.
