  We present the results of the search for gravitational waves (GWs) associated
with $\gamma$-ray bursts detected during the first observing run of the
Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO). We find no
evidence of a GW signal for any of the 41 $\gamma$-ray bursts for which LIGO
data are available with sufficient duration. For all $\gamma$-ray bursts, we
place lower bounds on the distance to the source using the optimistic
assumption that GWs with an energy of $10^{-2}M_\odot c^2$ were emitted within
the $16$-$500\,$Hz band, and we find a median 90% confidence limit of 71$\,$Mpc
at 150$\,$Hz. For the subset of 19 short/hard $\gamma$-ray bursts, we place
lower bounds on distance with a median 90% confidence limit of 90$\,$Mpc for
binary neutron star (BNS) coalescences, and 150 and 139$\,$Mpc for neutron
star-black hole coalescences with spins aligned to the orbital angular momentum
and in a generic configuration, respectively. These are the highest distance
limits ever achieved by GW searches. We also discuss in detail the results of
the search for GWs associated with GRB 150906B, an event that was localized by
the InterPlanetary Network near the local galaxy NGC 3313, which is at a
luminosity distance of 54$\,$Mpc ($z=0.0124$). Assuming the $\gamma$-ray
emission is beamed with a jet half-opening angle $\leq 30^{\circ}$, we exclude
a BNS and a neutron star-black hole in NGC 3313 as the progenitor of this event
with confidence $>99$%. Further, we exclude such progenitors up to a distance
of 102$\,$Mpc and 170$\,$Mpc, respectively.
