The collective magnetic excitations in the spin-orbit Mott insulator (Sr1−xLax)2IrO4 (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 Jeff=1/2 isospins, thus marking the departure of Sr2IrO4 from the essentially isotropic Heisenberg model appropriate for the superconducting cuprates.