Yes, iPod nano 1st Gen. I have this one. Same issue as OP, so I decided to change the battery.
iFixit‘s instructions are too complicated, however. I found somewhere in the Internet, YouTube I think, tutorials (of 3 to 4 minutes as far as I remember) that show easier ways, skipping some of the steps of iFixit‘s.
Actually, it worked, as my device now holds the charge for a while. It is two years ago, I think, that I did this. Then, there were sets with battery and tools available at Amazon. Dunno if this is still the case but you can of course buy at iFixit without doing every step of them
Most difficult part was the first: To get that thing opened...