  We present a thorough phenomenological analysis of the experimental data from
Belle Collaboration for the transverse $\Lambda$ and $\bar\Lambda$
polarisation, measured in $e^+e^-$ annihilation processes, for the case of
inclusive (plus a jet) and associated production with a light charged hadron.
This allows for the first ever extraction of the quark polarising fragmentation
function for a $\Lambda$ hyperon, a transverse momentum dependent distribution
giving the probability that an unpolarised quark fragments into a transversely
polarised spin-1/2 hadron.
