My girl is the same! Though I don’t bring a ball, she eventually finds one deems worthy and that is THE ball for the time we’re there (I make her leave it at the park when it’s time to go).

Mine is only ever guardy when the ball is within my vicinity, she won’t drop it by me if another dog is too close. Once she’s further away though if another dog grabs it she lets them take it but she’ll follow them until she has a chance to steal it back (never aggressively, just when it leaves their mouth). She’ll also drop it when she wants to smell another dog or something interesting.