My rabbit licks my hand too. It’s because they are trying to lick salt off of your hand. She probably is a teeny bit scared since she runs away as you said but most rabbits do that. All you have to do is carry on being gentle with her and talking to her softly. I find that lying down on my bed with my rabbit and letting her climb on me is good for bonding as long as you let THEM make the first moves. Gradually start petting her more and holding her after you feel she has your trust. Also, try hand feeding her. My rabbit HATES treats and I thought we were going to be hopeless then one day I tried her with a piece of Romaine lettuce and she loves it! Try to find (if you haven’t already) your rabbit’s favorite treat and hand feed her it everyday if its healthy, like mine.
As for the licking, just buy her a mineral stone. You can either get unflavored ones or flavored ones however the ones that are flavored usually have a bit of added sugars so be careful for that otherwise it might upset your rabbits tummy. My bunny doesn’t like the unflavored ones since they just taste and feel like chalk so I bought her one of the strawberry ones but it didn’t have a lot of sugar so she was good with it.

Bunnies don’t need salt blocks. They get vitamins and minerals from their pellets, greens, and hay. They also don’t like to be picked up — its instinct. They are prey animals so, by instinct, they are afraid that whatever picks them up is going to eat them. A rare bunny doesn’t mind being picked up, but the vast majority would much rather hop where they need to go on their own.

As for the licking, your bunny is grooming you! She’s telling you that she loves you.

If she’s licking you then she’s accepted you. She’d never accept a predator. One of my bunnies, Kieko (our single bun), is like that. When we pet her, she has to groom us back. She also grooms her stuffed bunny JackJack (she preferred to bond with him over our lops — refused to groom them, but we’re going to try again).

BTW, all three of mine will take meds from a syringe if we just hold the syringe down to them — they fight the meds like crazy if we pick them up to give meds. Worth a try if you ever have to give meds again.

She seems fine with it. Take a look at the July picture thread. I posted pics of her with JackJack. Maybe since he doesn’t groom her back she thinks he’s top bunny!