  A search for double $\beta$ decay of dysprosium was realized for the first
time with the help of an ultra low-background HP Ge $\gamma$ detector. After
2512 h of data taking with a 322 g sample of dysprosium oxide limits on double
beta processes in $^{156}$Dy and $^{158}$Dy have been established on the level
of $T_{1/2}\geq 10^{14}-10^{16}$ yr. Possible resonant double electron captures
in $^{156}$Dy and $^{158}$Dy were restricted on a similar level. As a
by-product of the experiment we have measured the radioactive contamination of
the Dy$_2$O$_3$ sample and set limits on the $\alpha$ decay of dysprosium
isotopes to the excited levels of daughter nuclei as $T_{1/2}\geq 10^{15} -
10^{17}$ yr.
