My daughter was just like this. She had a bladder of steel and would hold it in all day. I’d sit her on the potty or toilet regularly, keep topping up her drinks, offer all manner of bribes... and she would not do a wee. Wouldn’t wet herself either. She’d be in pain but she wouldn’t give in.
It drove me insane. Stopped and started loads and then gave up while I was pregnant with severe SPD and just couldn’t face trying anymore.
Eventually, at the grand old age of 3 years and 10 months, I told her she was going to start wearing knickers every day and that was that. Of course, it started predictably and she held it in all day.
Well, it got to bedtime and I lost my temper. I told her I didn’t care how long it took, we were going to stay in the bathroom until she did a wee. So she let out a small wee. First time ever. I’m sure she didn’t empty her bladder but it was a start.
Day two, she tried the refusal trick again and I said no, I’m not having that anymore, you WILL start weeing on the toilet/potty... and she did. Basically, I broke her iron will by getting angry 
It was plain sailing from there, no accidents ever. Poo on the toilet was a different kind of psychological warfare. This kid was something else 