ESP-01 does not run code after Deep Sleep
What sasso is describing is classic Deep Sleep Zombie mode. There's someone else here with ESP-01 boards with the same issue.  I'd bet you got your boards in blister packs from Inland.  I didn't see any ESP-01 boards from Inland at my local computer store, only ESP-01S, and that one works with Deep Sleep just fine.  I only bought the one for a test.  I believe it's the Flash chip, but we haven't proven that yet.  I have plenty of Winbond flash in case I find a board that goes zombie, and can prove or disprove my theory.  I have 8 different types of boards and modules, and all come out of Deep Sleep correctly.
edit: If the pic at Ali is correct, the Puya Flash shouldn't be causing an issue, presuming it's real and not a clone.  Half the time the pics are only 'representative examples' of what they ship.  The ESP8266 Arduino core has supported Puya for a while now.  Also, on sober reflection the Inland package I bought said ESP-01S, and it's actually an ESP-01 (false advertising).  I looked closer and didn't see the pull-up resistors. :-(
If you're powering it from that USB programmer board, it should work.  If you have a separate 3.3V supply, I'd be suspicious of the supply.  The current spikes as the ESP is booting will cause many switching supplies to drop below 2V briefly, and the ESP goes insane.  Without having a zombie in front of me, I'm at a loss to go a lot further as my 'scope leads aren't that long.