This anxiety is very common. In the best, most blameless organizational cultures, team members are allowed to say "I didn't get much done yesterday" on an occasional basis and are forgiven for it - but I realize that reality isn't often like that.
In reality, sometimes you have to just BS things, as others have suggested, by holding some work back from your big flow-state pushes, or by dithering around with words like "I looked into...". It sounds like your standups are more about making management feel better than they are about engineers removing obstacles from each other. That's a hard problem to solve and probably outside the scope of what you can affect. Just do the best you can to survive and be successful in the situation you're in.
If I had any advice to give, it's to learn from how harmful and toxic this pattern is, and as you move up into seniority and management, make efforts to not repeat this mistake. Standup should be a time to discuss how the TEAM is making progress on delivering value to customers, not a time to compare individual efforts against each other. Managers should be involved enough in the lives of their teams to be able to make disciplinary or promotional calls for individual contributors without standup being where they receive that signal.
Note that according to Scrum-in-theory the managers shouldn't even be at the daily standup. The entire purpose is coordination between the developers. You say "I am working on X" so that people who know or do something relevant to X are reminded to tell you about it (after the standup). Also, once in a while two people accidentally start working on the same thing (maybe the first one forgets to assign the ticket to themselves), and at the standup this becomes obvious.
However, Scrum-in-practice is usually something quite different.
In reality, sometimes you have to just BS things, as others have suggested, by holding some work back from your big flow-state pushes, or by dithering around with words like "I looked into...". It sounds like your standups are more about making management feel better than they are about engineers removing obstacles from each other. That's a hard problem to solve and probably outside the scope of what you can affect. Just do the best you can to survive and be successful in the situation you're in.
If I had any advice to give, it's to learn from how harmful and toxic this pattern is, and as you move up into seniority and management, make efforts to not repeat this mistake. Standup should be a time to discuss how the TEAM is making progress on delivering value to customers, not a time to compare individual efforts against each other. Managers should be involved enough in the lives of their teams to be able to make disciplinary or promotional calls for individual contributors without standup being where they receive that signal.