We present non-radiative, cosmological zoom-in simulations of galaxy-cluster formation with magnetic fields and (anisotropic) thermal conduction of one massive galaxy cluster with M vir ∼ 2 × 1015 M ⊙ at z ∼ 0. We run the cluster on three resolution levels (1×, 10×, 25×), starting with an effective mass resolution of 2 × 108 M ⊙, subsequently increasing the particle number to reach 4 × 106 M ⊙. The maximum spatial resolution obtained in the simulations is limited by the gravitational softening reaching ϵ = 1.0 kpc at the highest resolution level, allowing one to resolve the hierarchical assembly of the structures in fine detail. All simulations presented are carried out with the SPMHD code gadget3 with an updated SPMHD prescription. The primary focus of this paper is to investigate magnetic field amplification in the intracluster medium. We show that the main amplification mechanism is the small-scale turbulent dynamo in the limit of reconnection diffusion. In our two highest resolution models we start to resolve the magnetic field amplification driven by the dynamo and we explicitly quantify this with the magnetic power spectra and the curvature of the magnetic field lines, consistent with dynamo theory. Furthermore, we investigate the ∇ · B = 0 constraint within our simulations and show that we achieve comparable results to state-of-the-art AMR or moving-mesh techniques, used in codes such as enzo and arepo. Our results show for the first time in a cosmological simulation of a galaxy cluster that dynamo action can be resolved with modern numerical Lagrangian magnetohydrodynamic methods, a study that is currently missing in the literature.