The proton electric dipole moment (pEDM) experiment is proposed to be done with two counter-rotating beams in an all-electric storage ring. Radial magnetic field is one of the most critical systematic errors in the experiment as it mimics the EDM signal. Its secondary effect is a vertical split between the counter-rotating beams. According to the proposal of the pEDM experiment, that split should be monitored and compensated to picometer level by the beam position monitoring systems (BPM). We propose a BPM using Superconducting QUantum Interference Devices or SQUIDs to measure the beam splitting with such resolution that corresponds to magnetic field measurement with $10^{-18}$ T (aT) resolution. In this paper we experimentally demonstrate that the white noise of such magnetometers and SQUID read-out electronics can be decreased by more than two orders of magnitude down to 25 aT$\sqrt{\text{Hz}}$ by using 5 hour signal averaging. It demonstrates very high long-time stability and low intrinsic fluctuation level in the instrument. We expect that our next BPM version will be able to reach <2 aT $\sqrt{\text{Hz}}$ white noise level after 5 hour averaging. This noise level can be further decreased by using recently designed single-chip magnetometers with sub-micron size Josephson junctions.