An "alice band" is so named, after the one worn by Alice in Lewis Carroll's stories.
Is there a male equivalent, to describe those worn by men with long hair (e.g. David Beckham)?
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No; it's still called an alice band, or sometimes the gender-neutral hairband or headband. They might also be referred to as sport(s) bands but that term is slightly misleading as it could equally apply to a wristband or similar.
I wouldn't be surprised to see the tabloid press coin a term like "Beckham-band", but there's not a common male equivalent in widespread use.