Can you refuse to take first on a HBP?

[FPump]

The ASA Rule (8-1-F) says, "A batter becomes a batter-runner when a pitched ball not swung at nor called a strike touches any part of the batter's person or clothing."

A batter-runner is defined as a "player who has finished a turn at bat but has not yet been put out or touched first base".

So, the rule says that when a batter is hit, he becomes a batter-runner. A batter-runner is defined as someone who has finished a turn at bat. So, there isn't an option to turn down a base.

Ditto for all other rule sets.
As stated previously, a local league rule can be implemented to modify any rule in the sanctioning body's official rule book. This particular one (HBP) is modified often on the local house/rec league level....esp. for the lower age groups.
However, if your local league does not have such a rule, or if your team is playing in championship tournament play, there is no option. The batter must take 1B if hit by a pitch.
Until reading the OP, the possibility of this sitch happening (in championship/tourney play) never occurred to me.
If it ever did occur......
and the coach refused to have his/her runner take 1B after being HBP, and I could not convince the coach otherwise, I'd declare the batter OUT for "abandoning her effort ."
JMO