We report the ALMA observations of the most massive (star-forming) galaxy in the redshift range 3 < z < 4 within the whole Great Observatories Origins Deep Survey South field (GOODS-S) field. We detect a large elongated structure of molecular gas around the massive primeval galaxy, traced by the CO(4−3) emission, and extended over 40 kpc. We infer a mass of the large gaseous structure of Mgas ∼ 2–6 × 10$\mathrm{<mrow\; data-mjx-texclass="ORD">11</mrow>}$ M⊙. About 60 per cent of this mass is not directly associated with either the central galaxy or its two lower mass satellites. The CO extended structure is also detected in continuum thermal emission. The kinematics of the molecular gas shows the presence of different components, which cannot be ascribed to simple rotation. Furthermore, on even larger scales, we detect nine additional CO systems within a radius of 250 kpc from the massive galaxy and mostly distributed in the same direction as the CO elongated structure found in the central 40 kpc. The stacked images of these CO systems show detections in the thermal continuum and in the X-rays, suggesting that these systems are forming stars at a rate of 30–120 M⊙ yr$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. We suggest that the extended gas structure, combined with its kinematic properties, and the gas-rich star-forming systems detected on larger scales, are tracing the inner and densest regions of large-scale accreting streams, feeding the central massive galaxy. These results corroborate models of galaxy formation, in which accreting streams are clumpy and undergo some star formation (hence enriching the streams with metals) even before accreting onto the central galaxy.