  We present Spin-SILC, a new foreground component separation method that
accurately extracts the cosmic microwave background (CMB) polarisation $E$ and
$B$ modes from raw multifrequency Stokes $Q$ and $U$ measurements of the
microwave sky. Spin-SILC is an internal linear combination method that uses
spin wavelets to analyse the spin-2 polarisation signal $P = Q + iU$. The
wavelets are additionally directional (non-axisymmetric). This allows different
morphologies of signals to be separated and therefore the cleaning algorithm is
localised using an additional domain of information. The advantage of spin
wavelets over standard scalar wavelets is to simultaneously and
self-consistently probe scales and directions in the polarisation signal $P = Q
+ iU$ and in the underlying $E$ and $B$ modes, therefore providing the ability
to perform component separation and $E$-$B$ decomposition concurrently for the
first time. We test Spin-SILC on full-mission Planck simulations and data and
show the capacity to correctly recover the underlying cosmological $E$ and $B$
modes. We also demonstrate a strong consistency of our CMB maps with those
derived from existing component separation methods. Spin-SILC can be combined
with the pseudo- and pure $E$-$B$ spin wavelet estimators presented in a
companion paper to reliably extract the cosmological signal in the presence of
complicated sky cuts and noise. Therefore, it will provide a
computationally-efficient method to accurately extract the CMB $E$ and $B$
modes for future polarisation experiments.

Cosmology and Nongalactic Astrophysics;Instrumentation and Methods for Astrophysics