Cinny does this to me too, and if I approach her when she has decided she isn't going to come back inside (in the fenced back yard) she will even sort of "melt" down to the ground while looking away, and if I try to pick her up she shrieks and nips at me. I'm always very gentle with her, and I know she isn't afraid of me, she just doesn't want to come in till she's ready. The one thing I've seen that will sometimes lure her back willingly is a small piece of bacon. I use little pieces of bacon as treats to train her since she doesn't like regular dog treats. She used to be very afraid of being picked up so if I'm going to, I tell her first and go slowly.
If you are in a secure place, the best fix for this behaviour is to ignore the dog and leave, or pretend to leave. Get in your vehicle, perhaps. Then when they want in, make a show of eating something they like and keep them out for a bit. Attitude adjustment!
It is funny, but not so much when their safety relies on a prompt recall.
Usually they sit with their back to you when they give you the "cold shoulder". She is just not listening. We never put our dogs on their own unless they are fenced in. Our boy dog would go wild Basenji on us if he got free. Our current dogs are too separation anxious to leave us and roam too far. Really tasty food might help with recall but some basenji's love freedom more than food.
@mddebellis, I agree on the sitting with their back to you. Actually, it is possible to put a reliable recall on a Basenji, or at least on some Basenjis, but I would never rely on food because there is always going to be that time when the distraction is more compelling than the food. In my experience they usually want to stay in contact with you and don't stray too far, so in a relatively safe area off leash is doable, but the right distraction could cause them to take off and perhaps go far enough that finding their way back could prove difficult. That said, nothing is foolproof. Leashes break (or are bitten through!), get tugged out of someone's hands, etc. and I do find that a dog that is used to being off leash is safer when the unexpected happens, because they don't get all silly and start playing "keep away". So if possible it's good to give them some off leash experience, and work on a reliable recall, but not everyone has a good place to do that.
I do have a good place for off leash hiking and we do it daily with no problem. We have a mile of hiking trails we have made in our woods. She at times will charge after a rabbit but 90% of the time there is no problem. Today a neighbor was walking their dog on a nearby trail (which has only happened one other time in 3 years) and she looked up at me and then turned and ran toward them. There was nothing I could do but wait and she did return shortly. For a few days it will be leash only walking to make a point. But she ignores me when the desire is overwhelming. Fortunately this is a rare problem.
This area is at the edge of one of our local colleges, very little traffic, and about 40 acres of woods to explore. Doodle does great while we are in the woods! I filmed it because I knew what she would do. And I know why she does it... she just doesn't want to leave. She doesn't run away, but she will keep her distance and then gradually accept that the fun is over and we have to go home. I generally do not carry premium snacks with me, but I can trick her if I start walking back into the trails and she will come to continue on her walk. On this particular outing, she stepped on a sand spur "ouch!" and sat down until I could come rescue her. Sometimes she will just "give in" and come to the car. But most of the time, she pouts.