  We use the Fermi-LAT gamma-ray observatory to search for gamma-ray emission
from four nearby, debris disk-hosting main sequence stars: $\tau$ Ceti,
$\epsilon$ Eridani, Fomalhaut, and Vega. For three stars ($\tau$ Ceti,
Fomalhaut, and Vega), we establish upper limits that are consistent with
theoretical expectations. For $\epsilon$ Eridani, we find a possible spatially
coincident source with a soft energy spectrum of $dN/dE \sim E^{-3.6}$.
However, at this stage we are unable to rule out that this emission is due to a
more extended feature in the diffuse background. In the interpretation that the
emission is due to $\epsilon$ Eridani, the $> 100$ MeV gamma-ray luminosity is
$\sim 10^{27}$ erg/s $\simeq 3\times 10^{-7}$ L$_\odot$, which is $\sim
10^{10}$ times the gamma-ray luminosity from the disk of the quiet Sun. We find
$\lesssim 2 \sigma$ evidence of source variability over a $\sim 7$ year
timescale. In the interpretation that the gamma-ray emission from $\epsilon$
Eridani itself, we consider two possible models: 1) cosmic-ray collisions with
solid bodies in the debris disk which extends out $\sim$60 AU from the host
star, and 2) emission from the stellar activity. For the former model, assuming
a total disk mass consistent with infrared measurements, we find that the size
distribution of bodies is steeper than expected for a collisional cascade
state. If confirmed as being associated with $\epsilon$ Eridani, this would be
the first indication of gamma-ray emission from the vicinity of a main sequence
star other than the Sun.
