  The collective magnetic excitations in the spin-orbit Mott insulator
(Sr$_{1-x}$La$_x$)$_2$IrO$_4$ ($x=0,\,0.01,\,0.04,\, 0.1$) were investigated by
means of resonant inelastic x-ray scattering. We report significant magnon
energy gaps at both the crystallographic and antiferromagnetic zone centers at
all doping levels, along with a remarkably pronounced momentum-dependent
lifetime broadening. The spin-wave gap is accounted for by a significant
anisotropy in the interactions between $J_\text{eff}=1/2$ isospins, thus
marking the departure of Sr$_2$IrO$_4$ from the essentially isotropic
Heisenberg model appropriate for the superconducting cuprates.
