Can the word "that" be used to refer to people?

Solution 1:

Google ngrams shows "people who" being twice as common as "people that" around 1820, and increasing ever since while "people that" stays flat. "The man who/that" and "The person who/that" show similar patterns.

So, "who" is certainly more common than "that" in reference to people, and certainly there are authorities who say "that" is wrong, as you have seen.

They're welcome to. I will carry on speaking my own language and ignore "authorities" who have the temerity to tell me I am incompetent at it.

Edit: and do they seriously think there's something wrong with "Who is it that ... "?

Solution 2:

The NOAD has the following definition for that:

(plural that) [relative pronoun] used to introduce a defining or restrictive clause, especially one essential to identification.

• instead of which, who, or whom: the book that I've just written.
• instead of when after an expression of time: the year that Anna was born.

The Merriam-Webster's Collegiate has the following note about the usage of that:

That, which, who: In current usage that refers to persons or things, which chiefly to things and rarely to subhuman entities, who chiefly to persons and sometimes to animals. The notation that that should not be used to refer to persons is without foundation; such use is entirely standard.