Toddler too obsessed with newborn sibling

I was mentally prepared for my 3.5 year old daughter to be jealous of her new baby sister, but I was NOT prepared for the opposite. She is super loving and caring towards her baby sister.

But my issue is that she is too obsessed with her. Constantly trying to kiss her and play with her. Wakes her up from her nap to “play” etc

Baby girl spits up a lot of milk when my daughter bothers her a lot and is constantly in her face.

It all comes from a place of love which is why I don’t know how to fix this issue.

Any suggestions??