Estate Agent refuses to budge on price

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I need to sell my flat first so this is purely theoretical, but the agent called me and asked what price would I be willing to put forward for this flat I viewed over the weekend. I said that it would depend on what price my flat ultimately ended selling for and the estate agent said that there was no room to budge on price.

I queried this, and said that the flat had been on the market since April 2022. The estate agent said it had been under offer previously and the estate agent does not take properties off the market even when under offer (!)

I thought house prices were falling so this kind of response is a bit unexpected.

the estate agent said that there was no room to budge on price.

The owner has told their agent they won't budge.
The estate agent represents them. They pay for his services.
He passes on the seller's instructions to potential buyers. He's not answerable to you, you are not his client.

I queried this, and said that the flat had been on the market since April 2022. The estate agent said it had been under offer previously

Quite likely. Plenty of negotiations fall through, especially in the current climate.

and the estate agent does not take properties off the market even when under offer (!)

It's absolutely normal during sale negotiations, not to take down the agents sign or online property adverts. Just in case an ongoing offer all falls through, as above.

when i bought my flat, the owner took it off rightmove once he accepted an offer & I paid for and completed a survey on the property. At that time, most people thought this was a bit unreasonable!

I thought most owners would take the property off the market once the offer is accepted.

Maybe he thinks the vendor might switch to another agent so he wants to keep them on side by saying that he has a potential buyer willing to pay £X price but they just need to get their own property under offer first.
( I've always thought that 'taking it off the market' means noting the listing and the for sale sign as 'under offer' or 'SSTC', not removing the listing or sale sign altogether - surely agents want 'under offer' or 'SSTC ' to be very visible as it tells people that that agent gets properties sold).

For my current flat, they removed the listing altogether. I have seen other flats where they remove the listing altogether. In fact when i was a FTB, i was advised by others that i should make removal of the listing a condition of the offer. The seller said that he would remove the listing once the survey was done (to show that I was willing to make a financial commitment).

I thought most owners would take the property off the market once the offer is accepted.

Most sellers (and agents) will decline any new viewers while the agreed sale is in process; that's fair to avoid wasting anyones time. There's absolutely no guarantee an agreed sale won't be disrupted by surprise circumstances in either party , it's commoner than you may think.

I never count it a done deal until the money's changed hands.So the sensible thing is to keep the property in the public eye right up until completion.

Since you're in no position to offer let them sweat if they come back to you unprompted then you're probably the only potential buyer in the pipeline. Knowledge is power when you are buying and selling and not revealing your hand until you have to is all part of the game.

Could they have had an offer but hoping you would offer more? Though I guess they would just say that to give you a nudge.

The listing for our purchase was removed when we made our offer. I think I was given the same advice to make an offer that was subject to survey, contract and listing being removed which they did. Wanted less chance of being gazumped.

My colleague got a really low ball offer for his home and told the estate agent he wouldn't accept any future offers from that party who did come back and offer a lot more. He got significantly over asking but probably wasn't asking for too much in first place.