A search for $alpha$ decay of naturally occurring osmium isotopes to the lowest excited levels of daughter nuclei has been performed by using an ultra-low-background Broad-Energy Germanium $gamma$-detector with a volume of 112 cm$^3$ and an ultra-pure osmium sample with a mass of 118 g at the Gran Sasso National Laboratory of the INFN (Italy). The isotopic composition of the osmium sample has been measured with high precision using Negative Thermal Ionisation Mass Spectrometry. After 15851 h of data taking with the $gamma$-detector no effect has been detected, and lower limits on the $alpha$ decays were set at level of $lim T_{1/2}sim 10^{15}-10^{19}$ yr. The limits for the $alpha$ decays of $^{184}$Os and $^{186}$Os to the first excited levels of daughter nuclei, $T_{1/2}(^{184}$Os$)geq 6.8times10^{15}$ yr and $T_{1/2}(^{186}$Os$)geq3.3times10^{17}$ yr (at 90% C.L.), exceed the present theoretical estimates of the decays half-lives. For $^{189}$Os and $^{192}$Os also decays to the ground states of the daughter nuclei were searched for due to the instability of the daughter nuclides relative to $beta$ decay.
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