The BBC Northampton tweet which went up at 10:52 GMT read: "Breaking News: President Trump is injured in arm by gunfire #Inauguration."
The post was deleted shortly afterwards, but it had been retweeted.
A BBC spokesman confirmed the account had been hacked and said: "We are investigating and taking steps to ensure this does not happen again."
Replies to the original included: "Are you sure?" and "Seems odd that little old BBC Northampton would be first with this news."
A message posted on the Twitter account shortly after the initial Trump tweet read: "Message from OurMine: we detected unusual activity on this account, the account was hacked by someone and we are trying to fix the issue now."
The BBC asked US hacking group OurMine if it was responsible for the Trump tweet.
A spokesman said: "We didn't hack it at the first time. We saw unusual activity on the account and we re-hacked it to make sure if the account was hacked or not, and unfortunately it was hacked.
"We just tweeted that someone hacked it.
"We never hack anyone for no reason. We are a security group."
BBC Northampton posted a message apologising and saying: "We do appear to have been hacked and are looking into how."
In its statement the BBC said: "BBC Northampton's Twitter account was hacked this morning.
"The tweet was removed quickly, as soon as we realised the situation."
A BBC Twitter account which reported Donald Trump had been shot had been hacked, the corporation has said.