Mastercard, which stopped processing payments to the whistle-blowing site, said the attack had had "no impact" on people's ability to use their cards.
But the BBC has been contacted by a payment firm that said its customers had "a complete loss of service".
In particular, it said that an authentication service for online payments known as Mastercard's SecureCode, had been disrupted.
Other readers have also said that they have had problems with online payments. The scale of the problems is still unclear.
Mastercard has not responded to the claims.
Earlier, Doyel Maitra of the firm, said: "Mastercard is experiencing heavy traffic on its external corporate website - Mastercard.com - but this remains accessible.
"We are working to restore normal speed of service. There is no impact whatsoever on Mastercard or Maestro cardholders' ability to use their cards for secure transactions."
Anonymous, which claimed to have carried out the attack, is a loose-knit group of hacktivists, with links to the notorious message board 4chan.
It said that it has hit several targets, including the website of the prosecutors who are acting in a legal case against Wikileaks founder Julian Assange.
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Websites that are bowing down to government pressure have become targets”
End Quote Coldblood Anonymous
PayPal, which has stopped processing donations to Wikileaks, has also been targeted.
The firm said Wikileaks' account had violated its terms of services.
"On 27 November the State Department, the US government, basically wrote a letter [to Wikileaks] saying that [its] activities were deemed illegal in the United States," PayPal's Osama Bedier told the Le Web conference in France.
"And as a result our policy group had to make the decision of suspending their account.
"It's honestly, just pretty straight forward from our perspective and there's not much more to it than that," he said.
Other firms that have distanced themselves from the site have also been hit in the recent spate of attacks including the Swiss bank, PostFinance, which closed the account of Wikileaks founder Julian Assange.
The bank said Mr Assange had provided false information when opening his account.
Security experts said the sites had been targeted by a so-called distributed denial-of-service attack (DDoS), which swamp a site with so many page requests that it becomes overwhelmed and drops offline.
Paul Mutton of security firm Netcraft said that 1,600 computers were involved in flooding the site with spoof requests.
Access to Mastercard's site is still intermittent.
Noa Bar Yosef, a senior analyst at Imperva said the attacks were "very focused".
"It is recruiting people from within their own network. They are actually asking supporters to download a piece of code, the DDoSing malware, and upon a wake-up call the computer engages in the denial of service," he said.
Before the Mastercard attack, a member of Anonymous, who calls himself Coldblood, told the BBC that "multiple things" were being done to target companies that had stopped working with Wikileaks or which were perceived to have attacked the site.
"Websites that are bowing down to government pressure have become targets," he said.
"As an organisation we have always taken a strong stance on censorship and freedom of expression on the internet and come out against those who seek to destroy it by any means."
"We feel that Wikileaks has become more than just about leaking of documents, it has become a war ground, the people vs. the government," he said.
Some of the early DDoS hits failed to take sites offline, although that was not the point of the attacks, according to Coldblood.
"The idea is not to wipe them off but to give the companies a wake-up call," he said. "Companies will notice the increase in traffic and an increase in traffic means increase in costs associated with running a website."
DDoS attacks are illegal in many countries, including the UK.
Coldblood admitted that such attacks "may hurt people trying to get to these sites" but said it was "the only effective way to tell these companies that us, the people, are displeased".
Anonymous is also helping to create hundreds of mirror sites for Wikileaks, after its US domain name provider withdrew its services.
Coldblood said that the group was beginning to wind down the DDoS attacks so that it could concentrate on using "other methods which are more focused on supporting Wikileaks and making sure the Internet stays a free and open place".