  In this paper, we present new, unstable solutions, which we call quicksilver
solutions, of a $q$-difference Painlev\'e equation in the limit as the
independent variable approaches infinity. The specific equation we consider in
this paper is a discrete version of the first Painlev\'e equation ($q\Pon$),
whose phase space (space of initial values) is a rational surface of type
$A_7^{(1)}$. We describe four families of almost stationary behaviours, but
focus on the most complicated case, which is the vanishing solution. We derive
this solution's formal power series expansion, describe the growth of its
coefficients and show that, while the series is divergent, there exist true
analytic solutions asymptotic to such a series in a certain $q$-domain. The
method, while demonstrated for $q\Pon$, is also applicable to other
$q$-difference Painlev\'e equations.
