Synaptic devices with linear high-speed switching can accelerate learning in artificial neural networks (ANNs) embodied in hardware. Conventional resistive memories however suffer from high write noise and asymmetric conductance tuning, preventing parallel programming of ANN arrays. Electrochemical random-access memories (ECRAMs), where resistive switching occurs by ion insertion into a redox-active channel, aim to address these challenges due to their linear switching and low noise. ECRAMs using 2D materials and metal oxides however suffer from slow ion kinetics, whereas organic ECRAMs enable high-speed operation but face challenges toward on-chip integration due to poor temperature stability of polymers. Here, ECRAMs using 2D titanium carbide (Ti3C2Tx) MXene that combine the high speed of organics and the integration compatibility of inorganic materials in a single high-performance device are demonstrated. These ECRAMs combine the speed, linearity, write noise, switching energy, and endurance metrics essential for parallel acceleration of ANNs, and importantly, they are stable after heat treatment needed for back-end-of-line integration with Si electronics. The high speed and performance of these ECRAMs introduces MXenes, a large family of 2D carbides and nitrides with more than 30 stoichiometric compositions synthesized to date, as promising candidates for devices operating at the nexus of electrochemistry and electronics.