We present images of $eta$ Carinae in the recombination lines H30$alpha$ and He30$alpha$ and the underlying continuum with 50~mas resolution (110 AU), obtained with ALMA. For the first time, the 230 GHz continuum image is resolved into a compact core, coincident with the binary system position, and a weaker extended structure to the NW of the compact source. Iso-velocity images of the H30$alpha$ recombination line show at least 16 unresolved sources with velocities between -30 and -65 km s$^{-1}$ distributed within the continuum source. A NLTE model, with density and temperature of the order $10^7$ cm$^{-3}$ and $10^4$ K, reproduce both the observed H30$alpha$ line profiles and their underlying continuum flux densities. Three of these sources are identified with Weigelt blobs D, C and B; estimating their proper motions, we derive ejection times (in years) of 1952.6, 1957.1, and 1967.6, respectively, all of which are close to periastron passage. Weaker H30$alpha$ line emission is detected at higher positive and negative velocities, extending in the direction of the Homunculus axis. The He30$alpha$ recombination line is also detected with the same velocity of the narrow H30$alpha$ line. Finally, the close resemblance of the H30$alpha$ image with that of an emission line that was reported in the literature as HCO$^+$(4-3) led us to identify this line as H40$delta$ instead, an identification that is further supported by modeling results. Future observations will enable to determine the proper motions of all the compact sources discovered in the new high-angular resolution data of $eta$ Carinae.
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