The singular usage of "they" is supported by respected American and English dictionaries.
You could escape from this particular argument by saying, "No-one raised a hand."
However I see nothing wrong with saying, "No-one raised their hand". Merriam-Webster explicitly allows this usage.
2: his or her : HIS, HER, ITS —used with an indefinite third person
anyone in their senses — W. H. Auden
Merriam-Webster is probably the most respected dictionary of American English.
Note that W.H. Auden was highly literate, a poet, who died September 29, 1973, so we are not talking about recent or uneducated usage.
Note: @Mike Scott has pointed out in a comment that The use of “they” as a singular dates back to the 14th century. https://en.wikipedia.org/wiki/Singular_they#Older_usage
Here's the British usage from lexico.com, an offshoot of the famous Oxford Dictionaries.:
Belonging to or associated with a person of unspecified sex.
‘she heard someone blow their nose loudly’
‘Anyone who wants to voice their displeasure over that lot has to do a
lot of voicing.’