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We have studied the unusual time variability of an ultraluminous X-ray source in M 101, 4XMM J140314.2$+$541806 (henceforth, J1403), using Chandra and XMM-Newton data. Over the last two decades, J1403 has shown short-duration outbursts with an X-ray luminosity $\sim1-3 \times 10^{39}$ erg s$^{-1}$, and longer intervals at luminosities $\sim0.5-1 \times 10^{38}$ erg s$^{-1}$. The bimodal behaviour and fast outburst evolution (sometimes only a few days) are more consistent with an accretor/propeller scenario for a neutron star than with the canonical outburst cycles of stellar-mass black holes. If this scenario is correct, the luminosities in the accretor and propeller states suggest a fast spin ($P \approx$ 5 ms) and a low surface magnetic field ($B \sim 10^{10}$ G), despite our identification of J1403 as a high-mass X-ray binary. The most striking property of J1403 is the presence of strong $\sim$600-s quasi periodic oscillations (QPOs), mostly around frequencies of $\approx 1.3-1.8$ mHz, found at several epochs during the ultraluminous regime. We illustrate the properties of such QPOs, in particular their frequency and amplitude changes between and within observations, with a variety of techniques (Fast Fourier Transforms, Lomb-Scargle periodograms, weighted wavelet Z-transform analysis). The QPO frequency range $<$10 mHz is an almost unexplored regime in X-ray binaries and ultraluminous X-ray sources. We compare our findings with the (few) examples of very low frequency variability found in other accreting sources, and discuss possible explanations (Lense-Thirring precession of the inner flow or outflow; radiation pressure limit-cycle instability; marginally stable He burning on the neutron star surface).