  We study residual finiteness for cyclic central extensions of cocompact
arithmetic lattices $\Gamma < \mathrm{PU}(n,1)$ simple type. We prove that the
preimage of $\Gamma$ in any connected cover of $\mathrm{PU}(n,1)$, in
particular the universal cover, is residually finite. This follows from a more
general theorem on residual finiteness of extensions whose characteristic class
is contained in the span in $H^2(\Gamma, \mathbb{Z})$ of the Poincar\'e duals
to totally geodesic divisors on the ball quotient $\Gamma \backslash
\mathbb{B}^n$. For $n \ge 4$, if $\Gamma$ is a congruence lattice, we prove
residual finiteness of the central extension associated with any element of
$H^2(\Gamma, \mathbb{Z})$.
  Our main application is to existence of cyclic covers of ball quotients
branched over totally geodesic divisors. This gives examples of smooth
projective varieties admitting a metric of negative sectional curvature that
are not homotopy equivalent to a locally symmetric manifold. The existence of
such examples is new for all dimensions $n \ge 4$.
