If the blade penetrates between the metacarpal bones, and doesn’t displace them, then wrapping it will work as emergency first aid… provided she’s getting more help later on. Also, the knife, bandage, and possibly the wall, are all potential vectors for infection, so she’s going to need to treat her bandage with an antiseptic.
Generally speaking, ligaments connect bones to one another, while tendons connect muscles to bone. Severing or tearing either of those is a permanent injury that will require surgery to repair. Though, just taking a knife through the palm (between the metacarpal bones) won’t necessarily nick one.
If I recall correctly there is a string of tendons (juncturae tendinum) that bridge the second through fifth metacarpals for each finger, and that could be injured, but it might be possible to actually miss it, depending on the size and placement of the blade. It can also result in some truly disturbing looking injuries, if torn.
A tendon or ligament injury isn’t one you can just walk off. Anyone who’s experienced a sprained ankle first hand can tell you that. If they’re torn, you can’t “coax mobility” into it. This would be like trying to get a door working after you pulled the pins from the hinges. The pieces are still there, but they aren’t mechanically connected the way they need to be.
…seriously, don’t google these kinds of injuries unless you have a strong stomach or a high tolerance for really disturbing looking hands.
That said, the injuries that are likely to tear ligaments, would also probably amputate the finger, at least partially. So this is a somewhat less likely outcome, unless the blade connects across a knuckle. Fortunately, (thought, it might not feel like it at the time) fingers are one of those very rare cases where you’ve got convenient spares available.
With knife injuries, it’s usually a good idea to avoid moving and stressing the wound. It’s possible to cause more damage. This is also true of bone fractures (which is a real possibility if you’ve just taken a knife through the hand). Also, in a similar vein, if you’ve been stabbed, pulling the knife out is usually a bad idea, because it will further aggravate the wound and result in increased bleeding. (There are situations where this isn’t the case, such as where the blade is interfering with applying aid, but those are the exceptions, not the rule.)
Where I’ve experienced issues, fighting to get finger mobility back, has actually been in staff training. Blows to the knuckles are unavoidable. The staff creates a rough structure that (partially) protects against breaking your fingers when struck, but it does nothing for the bruising, which makes moving them after getting hit, much harder. Also it hurts like hell. Your hands provide absolutely critical tactile information to the brain, unfortunately, this remains true when the information being fed back is how much that hurt.
The stiffness lasts for a few minutes, and it’s actually something you can acclimate to from experience. The pain lasts a bit longer, I want to say 15-30 minutes, but I never actually timed it, and it’s been nearly 20 years since the last time I had someone clock my knuckles with a staff.
A similar injury could be suffered if your character is holding onto a ladder, pipe, or even a D-Cell Maglight, and takes a blow to the hand hard enough to raise a bruise.