Hi My Little Monsters
Yes! DD (now 5) used to do this quite a bit last yr
it always made me feel guilty (she's an only child) and a bit .. well ... crap ... for want of a better expression
some days she would do the usual running and throwing herself in to my arms routine and other days she would be screaming "Mummy I don't want to go home, I want to go to X's house" and hang on to a tree in the school play-ground - always in front of a row of assembled parents of course
The way I coped with it there and then was by saying "Well, we'll call X's Mummy when we get home" and by the time we got home, she'd have usually calmed down/forgotten about it - however - I did actually go ahead and arrange for some friends to come over after school once a week
It usually happened (looking back) when she was a bit upset/over-tired/over-excited and therefore finding the prospect of a transition from school to home too overwhelming
At 3, your dd is a bit young for play dates. Agree with others, distraction and bribery always good at this age!!
Oh and my dd has grown out of it (well, so far this term, she's been very happy to come home anyway!!) so hopefully it is just a phase .....