Context. The relationship between outflow launching and the formation of accretion disks around young stellar objects is still not entirely understood, which is why spectrally and spatially resolved observations are needed. Recently, the Atacama Large Millimetre/sub-millimetre Array (ALMA) carried out long-baseline observations towards a handful of young sources, revealing connections between outflows and the inner regions of disks. Aims. Here we aim to determine the small-scale kinematical and morphological properties of the outflow from the isolated protostar B335 for which no Keplerian disk has, so far, been observed on scales down to 10 au. Methods. We used ALMA in its longest-baseline configuration to observe emission from CO isotopologues, SiO, SO₂, and CH₃OH. The proximity of B335 provides a resolution of ~3 au (0.03′′). We also combined our long-baseline data with archival observations to produce a high-fidelity image covering scales up to 700 au (7′′). Results. ¹²CO has an X-shaped morphology with arms ~50 au in width that we associate with the walls of an outflow cavity, similar to what is observed on larger scales. Long-baseline continuum emission is confined to <7 au from the protostar, while short-baseline continuum emission follows the ¹²CO outflow and cavity walls. Methanol is detected within ~30 au of the protostar. SiO is also detected in the vicinity of the protostar, but extended along the outflow. Conclusions. The ¹²CO outflow does not show any clear signs of rotation at distances ≳30 au from the protostar. SiO traces the protostellar jet on small scales, but without obvious rotation. CH₃OH and SO₂ trace a region <16 au in diameter, centred on the continuum peak, which is clearly rotating. Using episodic, high-velocity, ¹²CO features, we estimate the launching radius of the outflow to be <0.1 au and dynamical timescales of the order of a few years.