  Corner percolation is a dependent bond percolation model on Z^2 introduced by
B\'alint T\'oth, in which each vertex has exactly two incident edges,
perpendicular to each other. G\'abor Pete has proven in 2008 that under the
maximal entropy probability measure, all connected components are finite cycles
almost surely. We consider here a regime where West and North directions are
preferred with probability p and q respectively, with (p,q) different from
(1/2,1/2). We prove that there exists almost surely an infinite number of
infinite connected components, which are in fact infinite paths. Furthermore,
they all have the same asymptotic slope (2q-1)/(1-2p).
