I full-time in my trailer without a propane heater.
I have two $20 ceramic heaters from Walmart (can get something very similar on Amazon though). Last winter I work camped at Amazon in Kansas and there were three nights when it got into the single digits. With the two heaters in 17' of RV and Reflectix up in the windows I could keep the living space inside about 50 degrees warmer than it was outside. When it got down to 10 outside that meant it was no warmer than 60 anywhere in the RV. Any colder than that and the plumbing running in my walls would start to freeze, there was no good way to heat inside those walls. Fortunately it never stayed in the single digits more than a night, things would thaw out the next day and I had no permanent damage.
So basically, yes you can full-time in cold weather without propane. It's not always very fun though. ;)