  We prove a p-adic Labesse-Langlands transfer from the group of units in a
definite quaternion algebra to its subgroup of norm one elements. More
precisely, given an eigenvariety for the first group, we show that there exists
an eigenvariety for the second group and a morphism between them that extends
the classical Langlands transfer. In order to find a suitable target
eigenvariety for the transfer we formalise a notion of Langlands compatibility
of tame levels. Proving the existence of Langlands compatible tame levels is
the key to pass from the classical transfer on the level of L-packets to a map
between classical points of eigenvarieties, which is then amenable for
interpolation to give the p-adic transfer.
