Context. The relationship between outflow launching and 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) has carried out long-baseline observations towards a handful of sources, revealing connections between outflows and the inner regions of disks. Aims. Here we aim to determine the small-scale kinematic 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 use ALMA in its longest-baseline configuration to observe emission from CO isotopologs, SiO, SO$_2$ and CH$_3$OH. The proximity of B335 provides a resolution of ~3 au (0.03). We also combine our long-baseline data with archival data to produce a high-fidelity image covering scales up to 700 au (7). Results. $^{12}$CO has a 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 of the protostar, while short-baseline continuum emission follows the $^{12}$CO 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 $^{12}$CO outflow shows no clear signs of rotation at distances $gtrsim$30 au from the protostar. SiO traces the protostellar jet on small scales, but without obvious rotation. CH$_3$OH and SO$_2$ trace a region <16 au in diameter, centred on the continuum peak, which is clearly rotating. Using episodic, high-velocity, $^{12}$CO features, we estimate the launching radius of the outflow to be <0.1 au and dynamical timescales on the order of a few years.
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