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We present new near-infrared photometric measurements of the core of the young massive cluster NGC 3603 obtained with extreme adaptive optics. The data were obtained with the SPHERE instrument mounted on ESO's Very Large Telescope, and cover three fields in the core of this cluster. We applied a correction for the effect of extinction to our data obtained in the J and K broadband filters and estimated the mass of detected sources inside the field of view of SPHERE/IRDIS, which is 13.5''x13.5''. We derived the mass function (MF) slope for each spectral band and field. The MF slope in the core is unusual compared to previous results based on HST and VLT observations. The average slope in the core is estimated as -1.06±0.26 for the main sequence stars with 3.5M☉ <M<120M☉. Thanks to the SPHERE extreme adaptive optics, 814 low-mass stars were detected to estimate the MF slope for the pre-main sequence stars with 0.6M☉ <M<3.5M☉, γ=-0.54±0.11 in the K-band images in two fields in the core of the cluster. For the first time, we derive the MF of the very core of the NGC 3603 young cluster for masses in the range 0.6-120M☉. Previous studies were either limited by crowding, lack of dynamic range, or a combination of both.