Hi there, we had the exact same thing with our DD who is now 5 but I remember it like it was yesterday. I really feel for you, I remember feeling like I was unable to function at all and with our daughter, she barely napped in the day either and I honestly don't know how I survived 🤣 with her though, she was generally fussy and we couldn't put her down without her crying and would only settle and sleep when she was either on me feeding (also EBF) or cuddling with DH. I'm not sure how you're finding BF but with us we knew there was an issue there which was adding to the problem as the latch was still very painful- but as she was gaining weight well, her nappies were fine and developmentally she was flying; HCP's did not take our struggle seriously. We sought help with a lactation consultant and eventually her latch improved and she started going a little longer in between feeds by about 10 weeks onwards. By about 3 months I think she was going about 2.5 hrs in between feeds and by about 6 months she could go 3 hrs. If you're not sure if she has a good latch or you think there might be BF issues, I would push for help there first as she may just be hungry.
I don't mean to make you feel worse however I will say that our daughter never really improved with naps during the day and didn't really go longer than 3hrs without waking until she was about 3 (I BF until she was 1 but she continued to wake at those intervals even without having any milk). She is now 5 and still does not sleep right through the night (she wakes once and comes into our bed). She is happy, healthy and very bright and we just learnt to eventually accept that she does not need much sleep and probably never will.
I also think that she never really learnt how to fall asleep herself as she used to fall asleep whilst BF so during her normal sleep cycles of waking through the night; she always needed either myself or DH to comfort her back to sleep. We knew this as the time and never figured out how to resolve the issue tbh because the only advise we ever got was either leave her to cry (which we didn't want to do) or put up with it, which is what we did, but with great difficulty.
We now have a 8 week old DS (also EBF) and I read a book called The No Cry Sleep Solution by Elizabeth Pantley which has some very gentle but effective ways in which to help your child sleep better but without any tears and I have been trying some of these techniques already with DS and started to notice little differences between him and our first (although difficult to do at the moment as he's suffering with colic). So that might be worth a read for you if you wanted some evidence based information about babies sleep and feeding. We also learnt via this book that our DD's 'bad sleep' wasn't actually as bad as we had thought and her behaviour was very normal for a young, BF baby- in spite of what others told us but comparisons were often made with formula fed babies which is just not the same (I have nothing against formula feeding btw, I'm just saying that the expectations around feeding and sleep are different). I am extremely glad I stuck to the BF but oh dear god the lack of sleep was tough 🤣 but I now believe there are ways to continue EBF without feeling so awful at the same time and I wish I had found a way to do this first time around- I am determined to do this with DS hence reading this book ha!
FYI nappies seem normal btw, I think anyway! Both DD and DS poop all day 🤣
Oh and have you tried BF lying down? You still can't sleep but this helped me feel like I had a little more rest through the night at least!
Good luck with it- you're doing amazing by sticking to the BF and I hope you find a way to get some more sleep soon.xx