  We study the dynamics of avalanche ionization of pure helium nanodroplets
activated by a weak extreme-ultraviolet (XUV) pulse and driven by an intense
near-infrared (NIR) pulse. In addition to a transient enhancement of ignition
of a nanoplasma at short delay times $\sim200$~fs, long-term activation of the
nanodroplets lasting up to a few nanoseconds is observed. Molecular dynamics
simulations suggest that the short-term activation is caused by the injection
of seed electrons into the droplets by XUV photoemission. Long-term activation
appears due to electrons remaining loosely bound to photoions which form stable
`snowball' structures in the droplets. Thus, we show that XUV irradiation can
induce long-lasting changes of the strong-field optical properties of
nanoparticles, potentially opening new routes to controlling
avalanche-ionization phenomena in nanostructures and condensed-phase systems.
