Can you do that, or do you need consideration. It makes sense that you can because one party does something to their detriment and the other party relies on that.
What consideration was given in exchange for keeping the offer open? If none, then “No”
Promissory Estoppel is where one party gives a promise which the promisee takes and changes their own position on the strength of that promise. Did the offeree ( in your hypothetical example ) change their position as a result of the promise to keep the offer open?