Context.Alongside the slow (s) and rapid (r) neutron capture processes, an intermediate neutron capture process (i-process) is thought to exist. It happens when protons are mixed in a convective helium-burning zone, and is referred to as proton ingestion event (PIE); however, the astrophysical site of thei-process is still a matter of debate. The asymptotic giant branch (AGB) phase of low-mass low-metallicity stars is among the promising sites in this regard.Aims.For the first time, we providei-process yields of a grid of AGB stars experiencing PIEs.Methods.We computed 12 models with initial masses of 1, 2, and 3M⊙and metallicities of [Fe/H] = −3.0, −2.5 −2.3, and −2.0, with the stellar evolution code STAREVOL. We used a nuclear network of 1160 species at maximum, coupled to the chemical transport equations. These simulations do not include any extra mixing process.Results.Proton ingestion takes place preferentially in low-mass and low-metallicity models, arising in six out of our 12 AGB models: the 1 M⊙models with [Fe/H] = −3, −3 andα-enhancement, −2.5, −2.3, and the 2 M⊙models with [Fe/H] = −3 and −2.5. These models experiencei-process nucleosynthesis characterized by neutron densities of ≃1014 − 1015cm−3. Depending on the PIE properties two different evolution paths follow: either the stellar envelope is quickly lost and no more thermal pulses develop or the AGB phase resumes with additional thermal pulses. This behaviour critically depends on the pulse number when the PIE occurs, the mass of the ingested protons, and the extent to which the pulse material is diluted in the convective envelope. We show that the surface enrichment after a PIE is a robust feature of our models and it persists under various convective assumptions. In ouri-process models, elements above iodine (Z = 53) are the most overproduced, particularly Xe, Yb, Ta, Pb, and Bi. Our 3M⊙models do not experience anyi-process, but instead go through a convectives-process in the thermal pulse with a clear signature on their yields.Conclusions.Thus, AGB stars at low-mass and low-metallicity are expected to contribute to the chemical evolution of heavy elements through thes- andi-processes. Our models can synthesise heavy elements up to Pb without any parametrized extra mixing process such as overshoot or inclusion of a13C-pocket. Nevertheless, it remains to be explored how thei-process depends on mixing processes, such as overshoot, thermohaline, or rotation.