  This article studies a linear scalar delay differential equation subject to
small multiplicative power tail L\'evy noise. We solve the first passage (the
Kramers) problem with probabilistic methods and discover an asymptotic loss of
memory in this non-Markovian system. Furthermore, the mean exit time increases
as the power of the small noise amplitude, whereas the pre-factor accounts for
memory effects. In particular, we discover a non-linear delay-induced exit
acceleration due to a non-normal growth phenomenon. Our results are illustrated
in the well-known linear delay oscillator driven by $\alpha$-stable L\'evy
flights.
