Hi OP, this happened to me.
I am in London, have been looking for an age, have a tiny budget, was priced out of one area after another, seemingly on a weekly basis, and could not see anything else on Rightmove I remotely liked, and they were all in horrible areas. So I sucked it up.
I had to pay 10k over asking to get the flat but it ticked a lot of boxes for me. I was comforted by the fact I wasn't the highest bidder, according to the estate agent, but that may have been a lie.
Having said that, I have thought estate agents were lying before, claiming there were offers from cash buyers and higher offers on other places I'd bid on... but, guess what, they weren't! And I missed out on one place after another because I was too suspicious and stingey to offer another few thousand!
The mortgage valuation valued the flat I'm (finally) buying at £5k less than asking price, so effectively I am paying £15k over the odds. I have a 40 per cent deposit so the down-valuation did not affect my mortgage offer.
I thought about going back to the vendor to negotiate a discount, but after listening to friends' advice, asking myself whether it was worth the grief of losing the flat for the sake of 2-3k, and the thought of starting the whole process again and possibly ending up buying a worse place for £10k more six months down the line, I thought better of it.
Now if the vendors try to stiff me for more money before exchange, which we are swiftly approaching, I will walk away from the sale. And leave bloody London.
One thing as estate agent said to me about property going for over the asking price by £10ks was "In London you'll always make the money back"... which over the long term (ie. 10 years plus) is probably true.
Good luck OP.