Context. Imaged in the gap of a transition disk and found at a separation of about 195 mas (~22 au) from its host star at a position angle of about 155°, PDS 70 b is the most robustly detected young planet to date. This system is therefore a unique laboratory for characterizing the properties of young planetary systems at the stage of their formation. Aims. We aim to trace direct and indirect imprints of PDS 70 b on the gas and dust emission of the circumstellar disk in order to study the properties of this ~5 Myr young planetary system. Methods. We obtained ALMA band 7 observations of PDS 70 in dust continuum and ¹²CO (3–2) and combined them with archival data. This resulted in an unprecedented angular resolution of about 70 mas (~8 au). Results. We derive an upper limit on circumplanetary material at the location of PDS 70 b of ~0.01 M_⊕ and find a highly structured circumstellar disk in both dust and gas. The outer dust ring peaks at 0.65′′ (74 au) and reveals a possible second unresolved peak at about 0.53′′ (60 au). The integrated intensity of CO also shows evidence of a depletion of emission at ~0.2′′ (23 au) with a width of ~0.1′′ (11 au). The gas kinematics show evidence of a deviation from Keplerian rotation inside ≲0.8′′ (91 au). This implies a pressure gradient that can account for the location of the dust ring well beyond the location of PDS 70 b. Farther in, we detect an inner disk that appears to be connected to the outer disk by a possible bridge feature in the northwest region in both gas and dust. We compare the observations to hydrodynamical simulations that include a planet with different masses that cover the estimated mass range that was previously derived from near-infrared photometry (~5–9 M_Jup). We find that even a planet with a mass of 10 M_Jup may not be sufficient to explain the extent of the wide gap, and an additional low-mass companion may be needed to account for the observed disk morphology.