Yes, and they also keep a record on WHO voted, (not how they voted except as you pointed out), as well. This information is made freely available to registered political parties.
Ballot papers (votes) and their corresponding number lists (matching voter to ballot paper) are kept for 12 months before destruction, and together allow who voted for who to be known - however - it seems no-one is allowed to inspect this except for the purposes of prosecuting an electoral fraud, so votes are to all intents and purposes private.
Perhaps the top downvote was for pedantry (?), but there is a difference between anonymous and private. The British system could issue ballots without recording to which specific person - that would be anonymous. You’d lose the ability to ask a particular voter if their counted ballot was the one they marked (substitution fraud), but I don’t think this ever actually happens.
