Galaxy clusters are being assembled today in the most energetic phase of hierarchical structure formation which manifests itself in powerful shocks that contribute to a substantial energy density of cosmic rays (CRs). Hence, clusters are expected to be luminous gamma-ray emitters since they also act as energy reservoirs for additional CR sources, such as active galactic nuclei and supernova-driven galactic winds. To detect the gamma-ray emission from CR interactions with the ambient cluster gas, we conducted the deepest to date observational campaign targeting a galaxy cluster at very high-energy gamma-rays and observed the Perseus cluster with the MAGIC Cherenkov telescopes for a total of ~85 h of effective observing time. This campaign resulted in the detection of the central radio galaxy NGC 1275 at energies E > 100 GeV with a very steep energy spectrum. Here, we restrict our analysis to energies E > 630 GeV and detect no significant gamma-ray excess. This constrains the average CR-to-thermal pressure ratio to be ≲1-2%, depending on assumptions and the model for CR emission. Comparing these gamma-ray upper limits to models inferred from cosmological cluster simulations that include CRs constrains the maximum CR acceleration efficiency at structure formation shocks to be <50%.Alternatively, this may argue for non-negligible CR transport processes such as CR streaming and diffusion into the outer cluster regions. Finally, we derive lower limits on the magnetic field distribution assuming that the Perseus radio mini-halo is generated by secondary electrons/positrons that are created in hadronic CR interactions: assuming a spectrum of E-2.2 around TeV energies as implied by cluster simulations, we limit the central magnetic field to be >4-9 μG, depending on the rate of decline of the magnetic field strength toward larger radii. This range is well below field strengths inferred from Faraday rotation measurements in cool cores. Hence, the hadronic model remains a plausible explanation of the Perseus radio mini-halo.