Can you please help me with advise on a situation to claim damages.

A person was selling some books with a notice to reply any offers within 7 days. I posted them a letter via first-class the next day after seeing the note, agreeing to buy the book for £100, or £4 more than any other bid, whichever was the greater. On the very same day I wrote them an email telling them that I confirm my posted letter re the book.

9 days after they posted the sales note, they delivered the books to another person who had made a bid £89. The seller confirmed they read my email but did not receive the letter by post.

The book's market value is £300, and I want to seek damages. Please give me some advise thanks.

Damages for what? Under standard common law principles, they never made you an offer. In fact they specifically solicited offers, one of which you made. I don't see any duty on their part to accept your offer, to accept the highest bid, or to do anything other than to reply to your offer within 7 days. If they failed to do this, by simply saying "no", how are you damaged? No offer having been accepted, there was no contract with you.

Incidentally, I have never really understood why some people send an e-mail and yet feel compelled to post the same thing. It's like wearing a belt and suspenders at the same time. :)