A 3D metal–organic framework (MOF) having single-molecule magnet (SMM) linkers was prepared in crystalline form by using a tetrairon(III) complex functionalised with two divergent pyridyl groups, namely [Fe4(pPy)2(dpm)6] (1; H3pPy=2-(hydroxymethyl)-2-(pyridin-4-yl)propane-1,3-diol, Hdpm=dipivaloylmethane). Reaction of 1 with silver(I) perchlorate afforded [Fe4(pPy)2(dpm)6]2AgClO4(2), which crystallises in a cubic face-centred lattice and exhibits two interlocked diamondoid networks. In 2, the SMMs act as linear ditopic synthons, and silver(I) ions as tetrahedral nodes coordinated by four pyridyl nitrogen atoms. The magnetic properties of 1 (S=5 and D≈−0.4 cm−1in the ground spin state) are largely preserved in 2, which shows slow magnetic relaxation with an anisotropy barrier of Ueff/kB=11.46(10) K in zero field and 14.25(8) K in an applied field of 1 kOe. However, crystal symmetry triggers highly noncollinear magnetic anisotropy contributions oriented at 109.47° from each other along the threefold axes of AgN4tetrahedra, a unique scenario fully confirmed by a single-crystal cantilever torque magnetometry investigation. Magnetisation curves down to 0.03 K demonstrated the occurrence of a wide hysteresis loop when the magnetic field was swept along one of the four Ag−N bonds. By symmetry, the crystalline compound can then be persistently magnetised parallel or antiparallel to the four main diagonals of the unit cell, although the crystals have no overall second-order anisotropy.