To be annoyed that someone will be sleeping in my bed for a week plus?

Also, as you are the carer for your mother, you surely can guide her, while she might want to appear to be the hostess with the mostest, it just isn't going to be possible to have her friend stay under your roof for more than the first and last days of her trip.

I apologise if someone has already pointed this out, but in her opening post, OP states that she pays half of all utility bills and buys all her own food.
Hardly rent-free!
And I'm with you OP in that you don't give up your bed and bedroom.
Your mum's friend stays in a hotel/Airbnb, or your mum/friend pays for you to stay in one, or a fold-up bed/airbed is bought and placed somewhere suitable for the friend.

How old are you?

How old is your Mum?

How old is the friend?

Have you ever lived independently?

Have you ever had a relationship?

Do you have any friends?

I'd approach the situation with this frame of mind - your mother's friend is welcome to stay for the first night of their visit to your area, and the last night, but everything in between, it really would be best if they found alternative accommodation.

If you can, do a quick search on what is available in your immediate locality by way of B&Bs and AirBnB's and the nearest hotel to you. So you have different price brackets for the friend to be able to stay in. Remind your mother that you need the bed/mattress you're on for your back and when said friend is gone home, you're expected to care for your mother and this will be considerably harder if your back is in bits.
Also, if your mother's friend stays elsewhere, then she is free to come and go as she pleases, do what she wants and also eat in the restaurant of the hotel or wherever she wants.

Have you used an inflatable bed recently? They are truly awful. Every time you move you will be woken by the ripple effect across the rest of the bed. A fold-out bed is a good idea, but perhaps the guest could use it. A bedroom is a very personal space. It seems unreasonable to have to give it up. I’d also say no more than 3 nights.

Absolutely not.
Completely unreasonable for you to have to give up your bed.
I wouldn't consider it.
She hasn't invited your mother to visit in 30 years?
She is lucky to be offered a blow up.
You certainly don't want a cheeky fxxker like her feeling too comfortable.
Cheap blow up or hotel.....her choice.

Can you message and say 'I am just trying to work out appointments and meals etc for time you are here. Could you give me your arrival and departure dates.'

She’s unwilling to say when she’ll be visiting. I didn’t know people got to dictate let alone pick and choose when they stay at someone’s home. She’ll just have to be told there’s no room or that she’ll have to sleep on the couch failing that book a hotel. You’re under no condition nor obligation to give your bed up.

A z- bed is an option if it's affordable
and there is room.

Have you even actually had a conversation with your mum about this?

The Z bed is the answer. Unless you pay top money, airbeds aren’t great and this woman sounds like she’s your mum’s age.

Your mum needs to grow up and stop being selfish if she can’t give up her falling in front of the TV for short visit.

Send her details of local hotels and travelodge places. It sounds like a one sided friendship. You can say you are both looking forward to seeing her, but naturally you are unable to provide a bed due to the serious health conditions you both have.

No way should you give up your bed and be in agony to meet her demands
Tell her now, that she is welcome( if she is) to visit but you are unable to accommodate her
If she wants to come, she'll need to find and pay for a hotel or b&b nearby
Do not back down.

Of course you shouldn't be expected to give up your bed! Your mum also has the right to invite a friend to stay with you. When I had an extra guest who I didn't have a bed for, I bought a futon, which cost me about £100, and which serves as a nice comfy chair in the bedroom when not in use.

It's not "mums house", it's both their house.

Just because the OP is not a child, it doesn't mean that it's not her home.

What a horrible way to put it "if you’re living rent free in someone’s home" it HER MUM! And as it happens, it's both their house anyway.

You sound awfully bitter and jealous.

But neither of the beds are available? So she’ll have to go to a hotel.

Your mum is possibly looking forward to her coming and it may give her a boost. I would ask your mum to buy a single bed and put that in her sitting room. A proper bed is better than a fold up or sofa bed. I know a random bed looks odd in a sitting room but who cares? Alternatively pop the single bed, if there's enough room in your mum's bedroom.

Has your mum actually asked you to give up your bed, or are you just assuming that she will expect it?

Erm no if she knows you she knows you have issues meaning you can’t just slum it on the sofa and will happily oblige to sleeping on the couch herself or a hotel.

The OP and her DM are each others carers, she pays half the bills and it’s her family home since childhood. She’s not temporarily staying there. She lives there. It is her home. I do t think it is as simple as Mum can turf her out her room any time she likes.

Ok I get you now, I wasn’t sure if you preferred the couch but as that’s not the case then yes I’d agree with you, you need to assert yourself. Why should you be uncomfortable in your own home? I understand making guests comfortable and welcome but she is your mums guest so why doesn’t she give up her bed and she sleeps on the sofa? If your mum won’t agree to that then surely she can see it is an unreasonable ask of you. Why can’t the friend sleep on the couch? I’d say to your mum no one is having your bed. The guests options are 1- a hotel 2- mums bed 3- the couch

She will not be sleeping in your bed. For heaven's sake any decent person would be mortified to inflict that on their host, and if she's not a decent person you shouldn't be hosting her at all.

Give the guest the sofa. Let her know in advance that whilst she's absolutely welcome, you don't have a guest room and can only offer a sofa. If she doesn't want the sofa she doesn't have to stay with you of course. Do not on any account follow your plan, it's a horrible thing to do to someone, because it tells them loud and clear that you think they are selfish.

But she can sleep on a hotel, right?

Errr.... yes. That is LITERALLY what hotels are there for 🙄

I'd definitely assert your right to keeping your bed. Your mother has to accept that part of the arrangement is that her grown daughter who lives with her as a carer and contributes to the costs of the running the house and has health issues that make sleeping on the sofa difficult cannot be turfed out of her own bed. That's insane. There are other options (airbed, hotel), and if her friend truly wants to visit, she can avail herself of one of them.