Oh OP. You are totally right that this isn't sustainable. I have a 3yo and a 6mo so I know what it's like but you really can't carry on like this. If you are actually banging your head on the wall, please tell your GP or HV. They will have heard it before and will be able to give advice.
Two questions: are you breastfeeding and will baby sleep in the cot / basket?
If you are breastfeeding, I get its much harder. Your DP needs to be helping by doing the nappy changes / resettling so you can get straight back to sleep. That will maximise your sleeping time. If you can express, you could always leave a bottle on a weekend overnight so that you can get a stretch from say midnight to 5am and he can do the feed in the middle.
If you are bottle feeding, then on a Friday or Sat night when your DP isn't working the next day, he needs to take responsibility for doing all the overnights to allow you to catch up with sleep. It's amazing what one or two good nights sleep will do to you.
If you dont have family around to help, if funds allow, you could also consider booking a night nanny for a night to allow you both to catch up. But they are pricey. A sleep consultant might also be able to offer some tailored advice.
Once you are caught up, you can think more logically and practically. We did the shift system that other pps have mentioned. I never thought i could sleep in the early part of the evening but blackout curtains, eye mask, white noise etc and you will drift off.
Will baby sleep in a cot / basket? Mine would only be held which made things harder but even if you can't sleep, when they are napping, make sure you are lying down in bed. You might find you just drift off for a bit. Look up Lullaby Trust safe sleeping if they won't go into the cot.
I also wonder if baby is overtired? Mine was sleeping about 13hrs a day at 3mo i think. You don't say if you are getting out and about but I would have a few days at home, let baby sleep whenever they want, don't wake them at any time and see what their natural rhythm is. My 6mo has only just settled into a nap routine so don't feel pressure to get into a routine too early.
Once baby is 6months, you can look at gentle sleep training / routine etc.
Its really important to look after yourself so well done on starting a thread for help.