Clusters of galaxies are expected to be reservoirs of cosmic rays (CRs) that should produce diffuse γ-ray emission due to their hadronic interactions with the intra-cluster medium. The nearby Perseus cool-core cluster, identified as the most promising target to search for such an emission, has been observed with the MAGIC telescopes at very-high energies (VHE, E ≳ 100 GeV) for a total of 253 h from 2009 to 2014. The active nuclei of NGC 1275, the central dominant galaxy of the cluster, and IC 310, lying at about 0.6° from the centre, have been detected as point-like VHE γ-ray emitters during the first phase of this campaign. We report an updated measurement of the NGC 1275 spectrum, which is described well by a power law with a photon index Γ = 3.6 ± 0.2stat ± 0.2syst between 90 GeV and 1200 GeV. We do not detect any diffuse γ-ray emission from the cluster and so set stringent constraints on its CR population. To bracket the uncertainties over the CR spatial and spectral distributions, we adopt different spatial templates and power-law spectral indexes α. For α = 2.2, the CR-to-thermal pressure within the cluster virial radius is constrained to be ≲ 1-2%, except if CRs can propagate out of the cluster core, generating a flatter radial distribution and releasing the CR-to-thermal pressure constraint to ≲ 20%. Assuming that the observed radio mini-halo of Perseus is generated by secondary electrons from CR hadronic interactions, we can derive lower limits on the central magnetic field, B0, that depend on the CR distribution. For α = 2.2, B0 ≳ 5-8 μG, which is below the ~25 μG inferred from Faraday rotation measurements, whereas for α ≲ 2.1, the hadronic interpretation of the diffuse radio emission contrasts with our γ-ray flux upper limits independently of the magnetic field strength.