“Guess who I met today?”

“Hmm… I_____(give up, gave up).”


I give up.
(present tense to refer to the action of giving up guessing right at that moment.)

Compare with:
She asked me to guess who she met, but after three attempts I gave up.

What about if you are asked to find something, but after looking all over the place for it, you still can’t find it. Then which one is used, [color=red]I give up or [color=red]I gave up?

By the way, [color=red]you are asked to find something or [color=red]you are asked to look for something?

The same applies - it depends on when it happens.
I’ve looked everywhere but can’t find it so I give up. can’t - present
I looked everywhere but couldn’t find it so I gave up. couldn’t - past

‘Find’ or ‘look for’ both work.

By the way, what is the bridge like thing that helps you cross over a busy street? And bridge-like or bridge like?

If I am right about what you are thinking, in the UK the ‘bridge-like thing’ would be a footbridge.