In this work, we present a highly retentive and synaptic-functional transistor memory device architecture based on the gate-deterministic remote doping of graphene via surface-oxidized Ti3C2TX MXene nano-floating-gates (NFG). Interestingly, unlike the conventional NFG-embedded architecture, the introduction of core/shell-like MXene under an electrolyte-gated graphene field-effect transistor (GFET) architecture induced a cooperative evolution of the hysteresis loop associated with ionic motion in the electrolyte gates and charge trapping/detrapping in the nanoflakes, resulting in a deterministic remote doping of the graphene layer. The resulting device exhibited a highly retentive memory behavior. In addition, synaptic functions having mechanical flexibility could be successfully emulated using MXene-based GFETs fabricated on a flexible polyethylene naphthalate substrate.