  We show that local-global compatibility (at split primes) away from $p$ holds
at all points of the $p$-adic eigenvariety of a definite $n$-variable unitary
group. The novelty is we allow non-classical points, possibly non-\'{e}tale
over weight space. More precisely we interpolate the local Langlands
correspondence for GL(n) across the eigenvariety by considering the fibers of
its defining coherent sheaf. We employ techniques of Scholze from his new
approach to the local Langlands conjecture.
