  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$)\geq 6.8\times10^{15}$ yr and
$T_{1/2}(^{186}$Os$)\geq3.3\times10^{17}$ 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.
