Using lithiums can overheat the flashlight, but the Peak Eiger will not electrically fail from the extra power. If you turn the Eiger on full blast with a li-ion in it and set it on the table, then the battery, electronics, or LED will self-destruct. If you use it in your hand, it will get uncomfortably warm to hold at full blast. The manufacturer's (admittedly terrible) site says 10440 is okay.
If you take responsibility for the fact that running a lithium-ion cell will overheat the light quickly on full blast, then use a li-ion carefully with crazy output. If not, stick to a NiMH AAA that removes the need to be careful. Note: If you plan to trust non-flashlight people with this light, they WILL turn it on full blast and put it down to work under their car. I had a ROP mod killed that way, "Hey this got really dim after a while but it was still bright enough to finish what I was doing."