If a private entity prohibits firearms and likewise provides "security"...?

has it entered into an implied warranty of safety with its customers? For example, if a mall prohibits its customers from entering while carrying legally-obtained and legally-carried firearms and simultaneously has "mall security", when a shooter enters that mall and starts murdering people, has the mall entered into a liability situation wherein it can now be held responsible in the civil courts for any deaths or injuries?

No.The implication is not a warranty of safety, but merely of reasonable measures to exercise a high degree of care to its invitee customers. And the hypothetical shooter is called an Intervening criminal act by a third party, for which the mall is not responsible. The only way the mall would be responsible would be if there were PRIOR assaults by criminals on the premises, and the mall FAILED to take any measures of protection. So by hiring guards, they are really mostly protecting themselves from tort or criminal liability.
The above is true regardless of the legality of owning or carrying firearms in the particular jurisdiction.

googo has it basically correct.

However, there was a rather goofy court decision in Michigan that determined an employee could sue for breach of contract for a contract between Ford (employer) and a company hired to provide plant security. The employee was shot and killed by a co-worker, and the court decided that the deceased employee was considered a third party beneficiary of Ford's security contract. (Komajda v. Wackenhut.)

So, can a court in your hypothetical find that customers are third party beneficiaries of the contract between the mall and its contract security firm? Who knows.

No.

About all I can add to the first answer is this:

Not just "no", but "hell, no."