  The production of ultra-pure raw material is a crucial step to ensure the
required background level in rare event searches. In this work, we establish an
innovative technique developed to produce high-purity (99.999%) granular zinc.
We demonstrate the effectiveness of the refining procedure by measuring the
internal contaminations of the purified zinc with a high-purity germanium
detector at the Laboratori Nazionali del Gran Sasso. The total activity of
cosmogenic activated nuclides is measured at the level of a few mBq/kg, as well
as limits on naturally occurring radionuclides are set to less than mBq/kg. The
excellent radiopurity of the zinc sample allows us to search for electron
capture with positron emission and neutrinoless double electron capture of
$^{64}$Zn, setting the currently most stringent lower limits on their
half-lives, $T_{1/2}^{\varepsilon\beta^+} > 2.7 \times 10^{21}\text{yr}$ (90%
C.I.), and $T_{1/2}^{0\nu2\varepsilon}> 2.6 \times 10^{21}\text{yr}$ (90%
C.I.), respectively.
