TMR-1 (IRAS~04361+2547) is a class~I proto-stellar source located in the nearby Taurus star-forming region. Its circumstellar environment is characterized by extended dust emission with complex structures and conspicuous filaments. A faint companion, called TMR-1C, located near the proto-star had been detected in previous studies, but its nature as a very young substellar object remained inconclusive. To improve the constraints on the nature of TMR-1C, and to investigate the process of very low-mass star formation in the TMR-1 system we use very sensitive infrared imaging observations as well as NIR spectroscopy. We construct the SED of TMR-1C over a much larger wavelength range as had been possible in previous work and compare it with models of extincted background stars, young sub-stellar objects, and very low-mass stars with circumstellar disk and envelope emission. We also search for additional low-luminosity objects in the immediate environment of the TMR-1, study the surrounding NIR dust morphology, and analyse the emission line spectrum of a filamentary structure in the physical context of a bow-shock model. We find that the observed SED of TMR-1C is inconsistent with an extincted background star, nor can be fitted with available models for a young extremely low-mass (<12M_Jup) object. Our near-IR spectrum indicates an effective temperature of at least ~3000K. Based on a good match of TMR-1Cs SED with radiation transfer models of young stellar objects with circumstellar disks, we propose that TMR-1C is most likely a very low-mass star with M~0.1-0.2M_sun surrounded by a circumstellar disk with high inclination, i>80deg. Moreover, we detect an additional very faint source, which we call TMR-1D, and that shows a quite striking symmetry in position with TMR-1C. TMR-1C and TMR-1D may have been formed from a common triggered star-formation event, caused by... (abstract abridged)
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