My daughter is nearly 5 and she still naps at least 5 days of the week. I distinctly recall around her 3s she started resisting the nap, but we stood firm regarding its importance. She was required to stay in her room, whether she slept or not. She usually slept, but not always. After a few frustrating days, as long as I was consistent with the requirement, she went back to napping regulary without a fuss. Then months later, the resistance would start again. It was as if she was testing each time.
So, now we're in a good routine where she will nap most days. And, since she is a bit older now, on the days she doesn't actually sleep, she will stay in her room ("quiet time") for at least 1 hour.
By the way, even on days she does end up falling asleep, she sometimes tells me: "Mommy, I'm not tired, I don't want to nap today." I usually reinforce the importance of naptime/rest for both of us. And I won't really offer her the choice of just doing quiet time in her room because she's likely to take it and not attempt to sleep. Also, on days she does not sleep she goes to bed 1 hr earlier than her usual time. That's not only necessary to ensure she gets enough rest, but it is also good because she sees it as the consequence of not napping.
Hang in there. Do what you feel is right for your little one. Hope this helps.