Those long afternoons were very dreadful for me initially. But it has been a while, and I've become so used to it. My sister who also has 2 girls told me, "Just pretend to fall asleep and she will sleep." But no, it didn't work at all. Mine would whine, cry and scream. My sister's coming-6 and 4 year olds are still taking afternoon naps. I've reasoned and threatened. Nothing works.
She'll always have her endless reasonings as well:
"but I'm not tired!"
"but I already have energy!"
"I want to go down the bed already"
"I'm awake already" (after closing her eyes for a few seconds)
"it's 4 o'clock already" (when i set a time restriction to lie on bed till a certain time)
"wake up!!!!!!" (when i pretended to sleep)
"I napped already. I closed my eyes when I was walking just now" (this is classic!)
For a short period of time, implementing quiet time worked. I made her do silent reading. But it only lasted a while.
As she approaches 4 years old, I've given up all hope that she would nap (yeah, I was still hoping!). She gets about 11 hours of night time sleep, and that seems sufficient for her. In the meantime, I'm still training myself to sleep earlier (I can't) so that I'll have energy to keep going the whole day.