  The $q$-Ornstein-Uhlenbeck processes, $q\in(-1,1)$, are a family of
stationary Markov processes that converge weakly to the standard
Ornstein-Uhlenbeck process as $q$ tends to 1. It has been noticed recently that
in terms of path properties, however, for each $q$ fixed the
$q$-Ornstein-Uhlenbeck process behaves qualitatively different from their
Gaussian counterpart in several aspects. Here, two limit theorems on the
extremes of $q$-Ornstein-Uhlenbeck processes are established. Both results are
based on the weak convergence of the tangent process at the lower boundary, a
positive self-similar Markov process little investigated so far in the
literature. The first result is the asymptotic excursion probability
established by the double-sum method, with an explicit formula for the Pickands
constant in this context. The second result is a Brown-Resnick-type limit
theorem on the minimum process of i.i.d. copies. With appropriate scalings in
both time and magnitude, a new semi-min-stable process arises in the limit.
