Context. The MIRI instrument on board JWST is now offering high-contrast imaging capacity at mid-IR wavelengths, thereby opening a completely new field of investigation for characterizing young exoplanetary systems. Aims. The multiplanet system HR 8799 is the first target observed with MIRIa s coronagraph as part of the MIRI-EC Guaranteed Time Observations (GTO) exoplanet program, launched in November 2022. We obtained deep observations in three coronagraphic filters, from a10 to 15 μm (F1065C, F1140C, F1550C), and one standard imaging filter at a20 μm (F2100W). The goal of this work is to extract photometry for the four planets and to detect and investigate the distribution of circumstellar dust. Methods. Using dedicated observations of a reference star, we tested several algorithms to subtract the stellar diffraction pattern, while preserving the fluxes of planets, which can be significantly affected by over-subtraction. To obtain correct measurements of the planeta s flux values, the attenuation by the coronagraphs as a function of their position must be accounted for, as well as an estimation of the normalisation with respect to the central star. We tested several procedures to derive averaged photometric values and error bars. Results. These observations have enabled us to obtain two main results. First, the four planets in the system are well recovered and we were able to compare their mid-IR fluxes, combined with near-IR flux values from the literature, to two exoplanet atmosphere models: ATMO and Exo-REM. As a main outcome, the MIRI photometric data points imply larger radii (1.04 or 1.17 RJ for planet b) and cooler temperatures (950 or 1000 K for planet b), especially for planet b, in better agreement with evolutionary models. Second, these JWST/MIRI coronagraphic data also deliver the first spatially resolved detection of the inner warm debris disk, the radius of which is constrained to about 15 au, with flux densities that are comparable to (but lower than) former unresolved spectroscopic measurements with Spitzer. Conclusions. The coronagraphs coming from MIRI ushers in a new vision of known exoplanetary systems that differs significantly from shorter wavelength, high-contrast images delivered by extreme adaptive optics from the ground. Inner dust belts and background galaxies become dominant at some mid-IR wavelengths, potentially causing confusion in detecting exoplanets. Future observing strategies and data reductions ought to take such features into account.