  It is well-known that Dark Matter (DM) inside a satellite galaxy orbiting a
host halo experiences a tidal potential. If DM is ultra-light, given its
wave-like nature, one expects it to tunnel out of the satellite - if this
happens sufficiently quickly, then the satellite will not survive over
cosmological timescales, severely constraining this dark matter model. In this
paper, we study the effects of the inevitable quartic self-interaction of
scalar Ultra-Light Dark Matter (ULDM) on the lifetimes of satellite galaxies by
looking for quasi-stationary solutions with outgoing wave boundary conditions.
For a satellite with some known core mass and orbital period, we find that,
attractive (repulsive) self-interactions decrease (increase) the rate of
tunnelling of DM out of it. In particular, for satellite galaxies with core
mass $\sim \mathcal{O}(10^7-10^8)\ M_\odot$ and orbital period $\sim
\mathcal{O}(1)\ \text{Gyr}$, one can impose constraints on the strength of
self-interactions as small as $\lambda\sim \mathcal{O}(10^{-92})$. For
instance, for ULDM mass $m = 10^{-22}\ \text{eV}$, the existence of the Fornax
dwarf galaxy necessitates attractive self-interactions with $\lambda \lesssim
-2.12 \times 10^{-91}$.
