Say that your sorry in a polite way (if he/she is angry about something that you did), point out you faults (always works!). You MUST swallow your pride
another thing that might work is let him/her in on a secret the he/she Will feel like a best friend to you (usually best friends let each other in on secrets) remember he/she isn't happy that he/she isn't talking to you, he/she just wont swallow his/her pride a give in. if YOU could give in and say "I was wrong, lets end this" then he/she will see the you are truly his/her best friend and also it will show that you are not a coward to giving in.
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A best friend that doesn't like you or talk to you might not be a good best friend to have. If something has recently happened to cause this, it's worth asking your friend to sit down quietly together to talk about what has happened and find a resolution. Perhaps that might be an apology from one or both of you. Perhaps it's a matter of just understanding each other's point of view about something. Maybe a promise to do something or not do something in the future could work.
If there is no reason for your friend not to like you or talk to you, it's time to look for a new best friend.
Give them some space and time to cool off, then try reaching out to them again to talk about what happened. Express that you value your friendship and are willing to work through any issues together. It's important to be patient and understanding of their feelings.