No - if Alice is selling in the course of business.
If Alice is not acting in the course of business, if she's a private seller, then Bob the buyer doesn't have the same statutory rights.
The Know Your Rights page in eBay.co.uk's Safety Centre outlines consumer rights law and the duties of businesses to consumers in the context of eBay.
The Distance Selling Regulations apply to fixed-price sales from business sellers including those on eBay.
The DSRs don't apply to auctions (s5(f) Excepted contracts).
The business and consumer cannot 'contract out' of the regs (s25 No contracting-out ).
In some cases business sellers on eBay have not abided by the law and sometimes eBay has not supported the consumer in such cases.The Electronic Commerce (EC Directive) Regulations 2002
Since late 2013 eBay.co.uk has required business sellers to inform consumers about the 14 day return period on fixed-price sales per the Consumer Contracts Regulations.Consumer Protection from Unfair Trading Regulations 2008
It is an offence for a business seller to misrepresent itself as a private seller.The Consumer Contracts (Information, Cancellation and Additional Charges) Regulations 2013
If Alice is a private seller the above regs do not apply.The Consumer Rights Act 2015
These laws apply to businesses selling to members of the public, they do not apply to members of the public selling to members of the public.
The Restrictive statements page provides more detail about the consumer's rights with regard to 'restrictive statements' such as "no refunds". That kind of statement is automatically void and it may constitute a criminal offence were Alice the business to give the consumer the impression they have less or no consumer rights to pursue a legitimate claim against her. Alice the business and buyer Bob cannot contract out of consumer rights law.
If Alice is selling in the course of business Alice must provide information about the business and the consumer's rights, including the cooling off or cancellation period. Alice commits an offence if she does not provide this information or if she misrepresents herself as a private seller.