We present high‐resolution simulations of a multiple merger of three disc galaxies, including the evolution of magnetic fields, performed with the N‐body/smoothed particle hydrodynamics (SPH) code Gadget. For the first time, we embed the galaxies in a magnetized, low‐density medium, thus modelling an ambient intergalactic medium (IGM). The simulations include radiative cooling and a model for star formation and supernova feedback. Magnetohydrodynamics is followed using the SPH method. The progenitor discs have initial magnetic seed fields in the range 10−9–10−6 G and the IGM has initial fields of 10−12–10−9 G. The simulations are compared to a run excluding magnetic fields. We show that the propagation of interaction‐driven shocks depends significantly on the initial magnetic field strength. The shocks propagate faster in simulations with stronger initial field, suggesting that the shocks are supported by magnetic pressure. The Mach numbers of the shocks range from approximately M= 1.5 for the non‐magnetized case up to M= 6 for the highest initial magnetization, resulting in higher temperatures of the shock‐heated IGM gas. The magnetic field in the system saturates rapidly after the mergers at ∼10−6 G within the galaxies and ∼10−8 G in the IGM independent of the initial value. These field strengths agree with observed values and correspond to the equipartition value of the magnetic pressure with the turbulent pressure in the system. We also present synthetic radio and polarization maps for different phases of the evolution, showing that shocks driven by the interaction produce a high amount of polarized emission. These idealized simulations indicate that magnetic fields play an important role for the hydrodynamics of the IGM during galactic interactions. We also show that even weak seed fields are efficiently strengthened during multiple galactic mergers. This interaction‐driven amplification might have been a key process for the magnetization of the Universe.