A population of globular clusters (GCs) has been recently established by the Fermi-LAT telescope as a new class of GeV γ-ray sources. Leptons accelerated to TeV energies, in the inner magnetospheres of MSPs or in their wind regions, should produce γ-rays through the inverse Compton scattering in the dense radiation field from the huge population of stars. We have conducted deep observations of the GC M15 with the MAGIC telescopes and used 165 h in order to search for γ-ray emission. A strong upper limit on the TeV γ-ray flux \lt 3.2× 10^{-13} cm^{-2 s^{-1}} above 300 GeV (\lt 0.26{{ per cent}} of the Crab nebula flux) has been obtained. We interpret this limit as a constraint on the efficiency of the acceleration of leptons in the magnetospheres of the MSPs. We constrain the injection rate of relativistic leptons, ηe, from the MSPs magnetospheres and their surrounding. We conclude that ηe must be lower than expected from the modelling of high-energy processes in MSP inner magnetospheres. For leptons accelerated with the power-law spectrum in the MSP wind regions, ηe is constrained to be much lower than derived for the wind regions around classical pulsars. These constraints are valid for the expected range of magnetic field strengths within the GC and for the range of likely energies of leptons injected from the inner magnetospheres, provided that the leptons are not removed from the GC very efficiently due to advection process. We discuss consequences of these constraints for the models of radiation processes around millisecond pulsars.