We perform general-relativistic magnetohydrodynamics zoom-in simulations of the merger of two neutron stars starting with pulsar-like realistic initial magnetic field. We employ unprecedented grid resolution down to $\Delta x\simeq 2.9$ m. We find rapid amplification of the magnetic energy at small scales, and demonstrate the action of a small-scale turbulent dynamo. We find evidence that ordered large-scale magnetic structures emerge already during the early post-merger evolution. These results indicate that strong magnetic fields can be established promptly after merger, with important consequences for the electromagnetic counterparts of binary neutron star mergers.