Wrist pain because of small bike frame?

It's not possible to say whether your bike fits from the information provided, but a too-small bike does sometimes end up with the bars too low, resulting in more pressure on the hand and wrist.

Bars also come with varying amounts of sweep which can make the angle of your wrist more or less natural. If your hand is not in a relatively neutral position when it's on the bars, you may put unwanted strain on your wrist.

Your question is best directed to a bike shop or fitter who can see you on the bike.

It could be for all sorts of reasons. And frame size is not necessarily the most likely of them.

I suggest you see a Physio for treatment and possibly (if you can afford it) have a bike fit by a trained fitter.

Are you placing alot of weight on your hands whilst cycling? Most of your weight should be on your sit bones, your hands are lightly placed on the handlebars. Higher handlebars will shift weight back onto pelvis.