More than 160,000 people were trying to access Currys PC World website at one point on Wednesday, which had only a limited supply.
The recommended retail price starts at £449 - but eBay bids are currently running to thousands of pounds.
Experts said a rise in bots buying consoles added to the high demand had led to technical difficulties.
PlayStation said the demand was unprecedented.
On Wednesday morning, a message on the Currys PC World site said it "now has limited PS5 stock".
But it sold out in under an hour.
And thousands were left in a virtual queue on the website throughout the day - with a warning message displayed to customers about heavy traffic, hours after the items sold.
"This morning, a limited number of PlayStation 5 [consoles] were made available online for purchase at the original RRP of £449," said Currys PC World gaming head Nick Hayden said.
"The website received a surge in traffic.
"And we sold out of stock within the hour.
"Due to very high demand for this product, we operated an online queuing system.
"We apologise if this caused any frustrations for our customers."