Can a beryllium 7 nuclei be stable if it has lost its electrons? Beryllium 7 decays via electron capture and I am (perhaps very naively) thinking that if it has no electrons, it will not be able to decay into lithium 7. Is it true? Or can the nucleus gain enough energy to go through beta+ decay?
 A: Investigations into changing the electronic environment of Beryllium-7 date back at least to a 1949 Physical Review paper where they hoped to measure a decay rate difference between Be metal and Be oxide. Given the equipment at that time they were unable to.
A more recent 1999 Earth and Planetary Science Letters paper measured the decay constants of $^{7}$Be in Be$^{2+}$(OH$_{2}$)$_{4}$, Be(OH)$_{2}$, and BeO. Quoting from their Discussion section:

In the present
study, by measuring the decay rate of $^{7}$Be in three
common forms of Be with an unprecedented high
precision of ~0.01%, it was shown that the half-life
of $^{7}$Be in natural environments could vary by as
much as 1.5%. The variation can be explained by a
change in electron density around the nucleus of Be
atom due to its association with different anions, and
hence different electronic polarizability and dipole
moments.

So, you don't even need to strip electrons from the atom, you only need to change the chemical environment (altering the charge distribution) to see an effect on electron capture.
