Context : Imaged in the gap of a transition disk and found at a separation of about 195 mas ( \sim 22 au ) from its host star at a position angle of about 155 ^ { \mathrm { o } } , 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 \sim 5 Myr young planetary system . Methods : We obtained ALMA band 7 observations of PDS 70 in dust continuum and ^ { 12 } CO ( 3 - 2 ) and combined them with archival data . This resulted in an unprecedented angular resolution of about 70 mas ( \sim 8 au ) . Results : We derive an upper limit on circumplanetary material at the location of PDS 70 b of \sim 0.01 M _ { \oplus } 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 \sim 0.2″ ( 23 au ) with a width of \sim 0.1″ ( 11 au ) . The gas kinematics show evidence of a deviation from Keplerian rotation inside \lesssim 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 ( \sim 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 . Conclusions :