YABU. Let's look at what you say: She has broken her finger at work, the finger is strapped to the next one and she has phoned in sick.
Firstly, if the finger is strapped to the next one, it is reasonable to assume she has had some sort of medical attention. She also knows the finger is broken. So again, one would assume she has had some form of medical advice from a qualified doctor. If she went to A&E or minor injuries unit, it would not be unusual for them to diagnose the injury, strap it up, advise her not to do x, y and z for a period of time, and tell her to self-cert in the first instance and then get a line from her GP if she needs more time.
Secondly - you make assertions about the extent of the injury however you are not medically qualified to make any kind of determination in that respect. I know from my job that broken fingers can turn very serious very quickly - it might sound like "just a sore finger" but the reality is different. Bear in mind how much we, as human beings, use our hands and fingers - even if the initial break isn't too bad, there is significant opportunity to do further and more serious damage to a broken finger if it is not supported and rested properly. I've seen numerous cases of seemingly simple finger breaks resulting in the person having a permanent disfigurement and loss of function of their finger due to it not being properly treated and/or cared for/rested. The only person who can advise this girl of what she can and cannot do is a qualified doctor - preferably a hand specialist. You cannot make that determination.
In terms of employment/you as a team leader, it is not only unreasonable but irresponsible and a potential breach of duty of care for you to try to push an employee to perform duties that could put their health and safety at risk. The girl has apparently injured her hand at work; notwithstanding that she now has 3 years to bring a claim against her employer if she so wishes, but if she returns to work and further injures herself, if she did not have a legal claim for the initial injury, she would most certainly would have one for any further subsequent injury, and that's before we start going into issues such as breaches of confidentiality, breach of the DPA, discrimination, bullying, harassment and employment tribunals.
I can tell by your posts OP what kind of "team leader" you are and let me tell you that a "leader" is the last thing you are. You are obviously a mircro manager on a power trip and I guarantee the people in your team are utterly miserable. Why do employers not understand that if they treat their employees like HUMAN BEINGS - with compassion, empathy and respect, and provide them with proper support, they will have a more loyal, hardworking and happy workforce who will probably stick around longer and be prepared to go the extra mile. It is proven time and again. As soon as employers micro-manage and have ridiculous sick leave policies etc... it makes employees miserable, demotivated, stressed and leads to employees working to rule. It is so simple yet employers still make the same mistakes over and over again. When I was off sick I tried to log on to do some bits of work and my employer called me and forbid me from working as they would be breaching their duty of care to me by having me work when I'm not fit to do so. I was paid sick pay, supported, shown empathy, respect... and when I returned to work I was happy to put in the extra hours to catch up and also help out my colleagues when they are off sick. It's called mutual respect and goes a long way in retaining good staff.
Treat your team member with respect and stop slagging her off on a public forum. If she has a broken finger then accept that there is little she can do in a supermarket without risking further injury to herself. Learn how to properly LEAD your team - stop micro managing. Staffing issues are YOUR responsibility and your employer's responsibility - they are not this girl's responsibility. Her only responsibility right now is to recovery from her injury.