A minor thing but

Yes the mud will dry and fall off but it will be all over your house. What a liberty!

The cheek of some people don't blame you for being annoyed ??

Advice from carpet salesman - let the mud dry first before trying to vacuum it up.

What is it with some dog owners! I wouldn't expect human visitors to tramp mud through the house so why is expected to be ok for dogs?
Don't get me wrong, most of my dog owning friends are pretty considerate but I don't own a dog for a reason

Reminds me of some visitors we had once - I said it was fine if they brought the dog and she could sleep in the kitchen or utility room. (our dogs had always slept in the kitchen).
First of all the dog galloped all over our wet vegetable patch, then tried to come into the sitting room covered in mud. I caught her and took her to the utility room to wipe her off with an old towel but got remarks about 'Aunty Jalima not letting you into the house, darling'. Then they fetched her in the middle of the night from the kitchen and let her sleep on their bed.

As predicted, when challenged about leaving a wet and dirty dog with me I was told that it would all fall off when it dried. To which I replied 'It has, in my house!' Followed up by asking how I was looking forward to visitors arriving this evening, so I pointed out that I needed to go and hoover.....again

What a cheek fetching a dirty and no doubt pungent dog to your home. Yes mud does fall off - all over your home.

We have a towel inside the backdoor to dry our dogs off after a walk.
My friend sometimes looks after our 3 dogs during the day if we are going somewhere that is not dog friendly and has had them for several days at a time on the rare occasion we stay in a hotel.
I would not dream of leaving my dogs if they were not clean and brushed, as after all she is doing me a favour and without her we would not be able to go away.
Maybe you need to make a fuss when your friend comes to collect her dog and say how long it took you to get the dog dry and clean and hope she takes the hint, or just tell her outright.