Castle of no Escape

This mostly required luck and persistence. If you take enemy damage just die and retry. I found the Bandit's dexterity helped when I was out of flares and landed on an enemy to get out without damage. If you luck out and get the lantern early it is much easier.
Make sure your dex is as high as possible to avoid damage at the boss. Not sure if it was necessary but, I backed out after each strike so the boss would miss me.

13 Jan 2021 Update

Following a further update from Xitilon this achievement has reverted to the previous criteria and as such below is no longer valid and LadyMagritte’s solution is now valid again.

***Previous Update Solution***

To get this to unlock you need to initiate a fight with a creature, hit it first and not kill it and evade its counter attack.

I’d evaded a lot of creatures before deciding to attack and my dexterity on the Bandit was high which helped evade.

Good luck and have fun.

Some tips that made this super easy for me:

Tip 1: From the character selection screen, if you push you start the game with a secret character with 20 for each stat. This is extremely helpful as most characters are strong in one stat, average in another, and weak in the third. The secret character however is strong in all three.

Tip 2: Abuse the web spell. When you come across a monster, press and then select the web spell. This prevents monsters from attacking meaning you can safely move off the square without taking damage. Plus you will find enough gold to replace intelligence you are losing by using the spells.

Tip 3: Do NOT open books and chests until you have the ruin that protects you from curses (I believe it looks like a yellow bookshelf). Curses of blindness are EXTREMELY annoying when trying not to take damage. Until you find the rune that protects you from curses, use the above tips to help you not take damage. Once you find the rune and no longer have to worry about curses open everything to get the best upgrades you can.

Tip 4: This should be obvious, but use the flares as much as possible by pushing on a non monster tile

Hopefully with these tips this achievement becomes easier.

While a lantern will definitely make the no-hit run a lot easier, using the IQ spell with the hidden character (press RT to use him, he start with 20 in each stat and has a max of 40) makes it a lot more possible.

This spell (used with RB) will instantly defeat any monster with INT less than or equal to yours. Otherwise, YOU instantly die. Once your IQ is over 20, the monster's IQ will be displayed when you are on their square. This means you know this very early in the run with the enchantress/hidden character, and never will know with the champion.

With a little bit of luck at the start getting flares and not getting hit when running away, you can upgrade your int to 40 and be able to take out most monsters you meet before they can hit you. Once you find the Sign and the Flame items (or else everything but the sign) you will be protected from all curses and can freely open chests and read books. (Chests may still explode but this will not void the achievement). I'd recommend waiting for at least the second weapon upgrade (the axe).

Once you find Nicolai you should have enough gold that you can keep him webbed while you hit him. Web him, hit him until the web goes away then web him again. It will cost you 1 INT each time. 40 INT should be more than enough, but if you start running low on INT, run away after webbing him and stock up again at the vendor.

I've compiled a list of the INT range I saw for each monster type across 10 encounters. It doesn't seem to be influenced by the floor, I saw mobs on the high end on floor 1, and the low end on floor 6. This way you know who you can safely use the spell on if your INT is too low to display the monster's INT, and even if it's high enough, you know which ones you are certain to beat before approaching them

<10 (This means any class can kill these monsters with the spell)
goblin(1-1)
imp(4-2)
skeleton(8-4)

<20 (Starting Enchantress or max INT champion can handle these)
rat(12-6)
snake(16-8)
gremlin(19-10)

<40 (Max enchantress can handle these).
kobold(24-12)
drider(32-17)
zombie(28-15)
hydral(34-18)
orc(40-22)

>40
vampire(41-23)
chimera(48-25)
encharmor(55-29)
ogre(52-44)
dragon(60-40)