Are there $\kappa$-closed $\kappa$-Aronszajn trees for $\kappa>\omega_1$?

Question

It is an easy exercise in Kunen that no pruned (i.e. without terminal nodes) $\kappa$-Aronszajn tree is $\kappa$-closed as a partial order (for $\kappa$ regular). I'm wondering if the same is true when we drop the restriction to pruned trees. That is: are there $\kappa$-closed $\kappa$-Aronszajn trees? It is straightforward to show that there aren't when $\kappa= \omega_1$, so my question is really: are there $\kappa$-closed $\kappa$-Aronszajn trees for $\kappa>\omega_1$? I'd also be happy with an answer relative to some extra assumptions like $GCH$.