We present a series of papers dedicated to modelling the accretion and differentiation of rocky planets that form by pebble accretion within the lifetime of the protoplanetary disc. In this first paper, we focus on how the accreted ice determines the distribution of iron between the mantle (oxidized FeO and FeO1.5) and the core (metallic Fe and FeS). We find that an initial primitive composition of ice-rich material leads, upon heating by the decay of 26Al, to extensive water flow and the formation of clay minerals inside planetesimals. Metallic iron dissolves in liquid water and precipitates as oxidized magnetite Fe3O4. Further heating by 26Al destabilizes the clay at a temperature of around 900 K. The released supercritical water ejects the entire water content from the planetesimal. Upon reaching the silicate melting temperature of 1700 K, planetesimals further differentiate into a core (made mainly of iron sulfide FeS) and a mantle with a high fraction of oxidized iron. We propose that the asteroid Vesta's significant FeO fraction in the mantle is a testimony of its original ice content. We consider Vesta to be a surviving member of the population of protoplanets from which Mars, Earth, and Venus grew by pebble accretion. We show that the increase in the core mass fraction and decrease in FeO contents with increasing planetary mass (in the sequence Vesta - Mars - Earth) is naturally explained by the growth of terrestrial planets outside of the water ice line through accretion of pebbles containing iron that was dominantly in metallic form with an intrinsically low oxidation degree.