Hello dianne243,
Thank you for using Apple Support Communities. It's my understanding that you're unable to successfully unlink contacts on your iPhone. You've come to the right place! I'm happy to help.
If you haven't already, restart your iPhone. Restarting can resolve many unexpected behaviors.
Restart your iPhone
Also, make sure that your iPhone has the latest software version.
Update your iPhone, iPad, or iPod touch
If the issue persists, delete the contact and add it again.
Manage and delete contacts on your iPhone, iPad, or iPod touch
Let us know the results. Please include any other troubleshooting steps you may have tried.
Have a nice day!