My partners cousin is obsessed with my baby

Trust your gut.You can never go wrong with it.You are her mum,if alarm bells are ringing,they are most likely ringing for a reason.
Your job is to protect your baby girl.You don't need to explain yourself to anyone(especially those blaming your hormones).

This could turn really serious. Do you need childcare? If not I'd just say oh no it's ok .... whatever reason.

If she asks to have her she's not well, she's teething, whatever it takes to get this person to back off. And tell her straight.

if she says anything like that again I'd just be blunt and say "look, I'm not trying to be funny but stop making these kind of comments, your making me uncomfortable. You are not her mum I am, ok. Now stop"

she has my LO quite a lot when I am at work or have breaks to see friends and so on.

Well you need to stop this immediately, don't you?

‘When you die she will be my baby’ wtf?!
This is outrageous and way out of order. It’s not banter and you’re not being over sensitive.
I echo what previous posters have said. Make other childcare arrangements. You can explain why, or not, it’s up to you.

This.

Obsessional behaviour. I'd be cautious

Missing not kissing.

I had to cut contact with a friend who was becoming very proprietorial over my daughter. She kept insisting we were family rather than friends, giving my daughter a different shortened version of her name and asking my husband to arrange video calls as she was kissing my daughter so much ( after I distanced myself). DH ignored the message.

due to working abroad and Covid etc she has met DD6 maybe ten times in total and is not a significant person for Dd6. She invited DD6 out for the day and DD6 was categorical she didn’t want to go without me. But I have two other close friends who DD goes off on little adventures or out to dinner and doesn’t even wave goodbye as she heads out of the door.

when we had our daughter and visited another friend and her family, it transpired that the friend was struggling with secondary infertility and during the overnight stay said that they would like to be guardians if we both died ( already have one in place) and that her son really wanted my daughter as a sister and said to her son in front of us “we could just keep her couldn’t we and they could have another baby” and repeated this in a text.

we kept our distance and saw them again when DD was three years old- the definitely didn’t want her then 😂😂

Do you have a will that says what would happen to DC if you and DH do actually die? I’d sort that out if not, just in case!
And then I’d just choose some stock phrases to repeat in those situations such as “No, she’s with Grandma today. Aren’t we lucky to have so many different people around her?”

She sounds very young. Those sort of comments usually come from teenage girls who are baby-obsessed but they outgrow it.

Perhaps don't make use of her baby-minding services too often from now on.

Or just stop telling her (and others?) things so she doesn’t know what you’re doing and when you need childcare and who is providing childcare. If she asks questions just be direct and tell her you’ve made a rule that you’ll not discuss these things with her as she tries to prevent others having time with DD.

just be clear and boundaried.

She's not a stranger so I doubt anything bad will happen.
However, don't allow your daughter to stay there over night, always pay her, thank her and call her a great babysitter.

Limit the times she babysits to a sensible no. of times per month and don't take advantage of her offers.
Suggest her as a babysitter to another of your friends.
Label her a babysitter and expect that she will act like one.

Don't inform her when you ask other family members to look after DD. Keep that private.

She adores your child. Does your DD enjoy spending time with her? If so, it's a plus plus.

The godmother should be someone of your choice but it wouldn't hurt for your DD to have two, or a couple as Godparents and her older adoring cousin..

Each time she creates drama tell her that her behaviour is not acceptable and stop contact for a couple of weeks. Tell her that you will not be manipulated and there will be consequences each time she try’s to prevent others spending time with DD.

She is your child, not the OH's families. You call the shots. You don't owe them an explanation.

Seems like you have weak boundaries. Please sort proper childcare, esp for work. Seems like she has your DD an awful lot. Why is that? You cannot have things both ways.

Is there any chance she is neurodivergent?
Another take is that she is young, immature, trying to show you she loves your child and takes the responsibility seriously, but doesn't have the social skills to know how to say this. Thinks she's saying it as a wry, edgy but loving joke and not able to pick up the signs that she's massively overstepping.

I wouldn't use her as any form of childcare at all, however frequently or infrequently.

Stick to your guns. Her behaviour is over the top and weird. So what if your partner's family criticise or patronise you about your stance! Either just ignore them or tell them that you are the parent so will do as you see fit in the face of this odd behaviour which you don't like. Your baby, your rules! The end.

Whatever it takes keep that woman away from your baby.
This all sounds very concerning, I wouldn’t leave a baby with her for any length of time. She sounds unsound and capable of anything.

I think you need to quantity 'quite a lot'. It sounds as if she is more or less your child minder and, as a result has grown very close to your baby.
Are you paying her or is she providing this service for free?
Unfortunately, it sounds as this situation has been ongoing for some time and whilst clearly unacceptable it is now normality.
Maybe she just wants to prove to you and everyone in the family how good and invested she is in looking after your baby.
To cut contact would be unfair on all involved including your baby but maybe try and cut down on using her for childcare, paid or otherwise.

Just be so ‘busy’ she can’t come and babysit, if she messages leave her hanging for a few days. She can’t force herself on you to see your baby. You need to stop her being around your baby or this will continue.

she doesn't sound well in the head. and I say that as someone with mental health issues myself, carer to a schizophrenic husband, and plenty of mental health issues in my family.
the things she says are not normal things to say.

Absolutely not. This is creepy. I had a baby sitter like this and could not fire her fast enough.

Anybody would find all this deeply concerning. I mean I find it concerning and I'm so old it would take lashings of HRT to make me hormonal. It's your OH's cousin. She is not even your baby's cousin and she's making outrageous statements about it being her baby if you died. I mean how can your OH not see how deranged or fixated his cousin is?

I wouldn't care what her family thought about not being left alone with her. Tell them that she's got so fixated on the baby that it's unhealthy. What 19 year old woman spends all her spare time with a baby who's a very distant relative and goes around saying if the baby's mother dies it will be her baby - conveniently ignoring that she's a teenager living at home and that the baby has a father too. I would be banning her from the house and rethinking your childcare arrangements. Your OH should be supporting you in this. You should both be putting your child's safety before his cousin's feelings. She sounds deeply irrational.

Is it realistic for your DH to work full time and then run your business in the evenings? Wouldn't it make more sense for him to look after the baby in the evenings? Of else have more time in the nursery with you working in the daytime?