Do you have any friends who have calm nonlicky, very obedient dogs?
Start of having conversations about the dog in her earshot. Act natural, don't include her in the conversation, just make sure she can hear. Funny stories, comments how sweet he is.
The arrange to run into them at the park. Don't tell her beforehand. When the hysterics start, tell her gently but firmly to calm down, its ok, he's not coming near you if you dont want him to. Resist the urge to pick up. Its really hard but don't reinforce the fear by 'rescuing her.'
Mummy needs to speak to Wendy, wait over there on the swing, away from the dog. Take her there if needed. Have a brief, 60 second chat with Wendy, with the dog sitting quietly beside you, pat it on the head and part ways smiling.
Continue about your day with no comment.
Arrange a few more chance meetings with the dog. Get her used to the idea that she is in control and can choose to wait at a distance without the dog coming over or being forced to interact.
Eventually after several months of these run ins, try to get her waiting nearer. Start meeting places where she can only really get a metre or two away and let her realise that the dog has been told to sit and will obey and isn't interested in her.
Again after a long while, once she can stand close by without panicking and a degree of trust has been achieved, engineer a situation in which Wendy needs you to hold the lead for a second. Hand him back without comment and stroke him on the head, tell him good boy. Let her see you interacting positively with him.
Eventually you would ideally work up to her holding the lead for a second and praising him and eventually to touching him but right now just focus on the standing nearby without meltdown stage and showing her positive interactions between you and the dog.
As for the fear of being licked, I have theory. People often use the phrase 'don't worry, he's friendly, worst he'll do is lick you to death.' Has she heard this and taken it literally?
Maybe would be worth letting her see you petting and being licked by a friendly dog. Let her see that nothing happens, it's just a bit gross.