We present an extensive analysis of the trapping processes induced by drain bias stress in AlGaN/GaN highelectron-mobility transistors (HEMTs) with p-GaN gate. We demonstrate that: (i) with increasing drain stress, pulsed I-V and VTH measurements shown an initial positive VTH variation and an increase in RON then, for drain voltages >100 V, VTH is stable and the RON shows a partial recovery. (ii) At moderate voltages, VTH instability is related to trapping at the gate stack, due to residual negative charge left behind by the holes that leave the p-GaN layer through the Schottky gate contact and/or to trapping at the barrier. At higher voltages, we demonstrate the interplay of two trapping processes by C-V and pulsed drain current analysis: (iii) a fast storage of positive charge, accumulated near the buffer/SRL interface, not strongly thermally activated, dominating at higher voltages; (iv) a slower negative charge storage, thermally activated with activation energies for trapping and de-trapping equal to ~0.6 eV and ~0.4-0.8 eV, respectively.