Good question, I was contracted in 2012 specifically to help build EMP protection for the military in southern California. I can confirm that lead acid cell batteries are resistant to EMP's because the electrodes are large enough to conduct most currents produced.

Most (about half produced) small batteries are not. These are batteries like AA's and AAA's because most of those batteries have zinc powder and this can move through the battery during an EMP and short out the cells.

However NiMh/NiCd/LiIon batteries in most electronic devices and usually have tiered foil construction and this design could cause the battery to explode! Think Samsung only everywhere.

You're better off with the a lead acid/wet cell batteries.

...And to be completely honest, your battery powered device not working during or after an EMP would be the least of the problems.

Thanks for reading!