I have characters with ice powers. I was planning that they could freeze an injury and prevent it from progressing. An arrow is shot at their heart, but I could possibly change the location to make the situation more probable.
Unless you use a healthy dose of story telling Magic, No.
The heart is constantly moving and pumping blood through out your body. Freezing something that needs to move isn't going to work. It also doesn't help that your blood will likely freeze, causing the water in it to expand into ice crystals which will tear up your blood vessels as your heart tries to pump it through your system. You would also be dropping your core body temperature since you are literally freezing your heart.
So realistically. No.
But with good story telling, you never know. Maybe the arrow just missed the heart, you could certainly use some ice to freeze it in place to make sure it doesn't shift around and cause more harm. Or maybe you created a tiny ice shield in front of your heart that stopped the arrow from piercing it (which you then quickly removed).
High heat can cauterize (think "weld") a wound. Ice can only immobilize — at a price
Any temperature outside the range of what the body wants to experience is damaging. To much heat and you burn. Too much cold... well, we call that "burning," too (freezer burn). But where heat will solidify the fluids around a wound (cauterization) such that the body can return to room temperature and not bleed, cold cannot.
And it's that solidification process that's the problem. sure, enough heat (or heat over a long enough period of time) will solidify the flesh, too (known as "cooking"). But cold doesn't have the benefit of staying solid at room temperature, which means it must be kept cold, and all the surrounding flesh with it. And it doesn't help that the body generates its own heat, which is constantly in conflict with the ice, so you must constantly keep reinforcing the ice.
So, staunching a wound is kinda out of the question (unless you can be really, really, really precise with the application of the ice!). But, what could you do? Frankly, anything a split can do. For example, if you're dealing with a broken leg, an ice cast could be formed around the leg (especially if clothing is left in place) to immobilize the leg.
You can also assist with medical treatment. For example, it's common to lower body temperature during cardiac surgery. Ice would be a dramatic but useful tool for that purpose (if carefully controlled, of course).
But, to help with, say, a knife wound? Not really an option. Think about how it would feel if the hot knife used in an emergency to cauterize a wound was left on the body and always had the same, heated temperature. The damage and pain would be amazing. Ice would have the same problem.
If you want to use ice powers to stop a wound, freezing the wound isn't an option. But cryopreservation is. Freezing wounds isn't really a good method for healing them, but it is a good method to stop cellular decay. If someone suffers a fatal wound, immediately dropping their body tempature to around 35 degrees Farenheit will be helpful. Sure, it kills the man, but it delay things like the four-minute mark to permanent brain damage.
Lower than that is freezing. And once you freeze cells, that's a lot of completely irreversible damage. That said, the wounded person is still dead once you drop the core tempature that low, because the human body wasn't meant to come back from tempatures that low.
On the flip side, if there's healing magic (or, as per Clarke's Third Law, technology) that can regenerate limbs, or something of that nature, than a person deep on ice like that can be revived, because the brain isn't dead.
Maybe, depending on how your magic system works. If you're literally using ice, probably not. However, if you're using more metaphorical associations of ice to conjure up some form of slowness or stasis magic, then it might be possible to do so. You might also be able to use ice's associations with slipperiness to make the arrow glance off.
Ice, no. Not if we're talking about big lumps of frozen tissue. But if your character has very fine control over their power, then we can work with this.
Perhaps they can rapidly chill the tissue, to a point just below zero? If you actually get ice crystals forming, you're going to have widespread tissue damage (as other commenters have noted).
Can the character direct this chilling power very specifically? Can it be directed to individual vessels (that would constrict with the cold)? Thus a wound to a major organ or limb could be effectively shut off (effectively acting as an internal tourniquet). If you go with this route, I'd recommend you go with the liver. Very highly vascular, will bleed a lot and cause a life-threatening injury, but can survive cold and has great regenerative capacity.