While i don't think that this company has the highest quality faucets, they are ok and do seem to stand behind their product. My daughter had one installed 18 years ago which was just replaced for free because of a defect. Her new faucet came with the pop up drain and no hole on the faucet.

Look at the vsrious styles here Price printer faucets

Something like this? https://www.homedepot.com/p/KOHLER-Cursiva-Single-Hole-Single-Handle-Bathroom-Faucet-in-Polished-Chrome-K-R30577-4D-CP/312772675

The spec mentions a "clicker drain" (which I think is the same as a push-pop drain), and claims "drain kit included"

Some more KOHLER options (all mention "clicker drain" and "drain kit included"):

https://www.homedepot.com/p/KOHLER-Rubicon-4-in-Centerset-Single-Handle-Bathroom-Faucet-in-Polished-Chrome-R29402-4D-CP/323706198

https://www.homedepot.com/p/KOHLER-Sundae-Single-Handle-Single-Hole-Bathroom-Faucet-in-Polished-Chrome-K-R28795-4D-CP/318241313

https://www.homedepot.com/p/KOHLER-Numista-Single-Handle-Single-Hole-Bathroom-Faucet-in-Polished-Chrome-K-R26583-4D-CP/317026158

Brizo has many if you're willing to spend the money.

1. Browse here: Single-hole chrome faucets with drain assembly.
2. Find faucets you like.
3. Check the description for "pop-up drain assembly".
4. To confirm, open the faucet webpage and check the specifications and exploded parts lists.

Example: Moen Hilliard 1.2 GPM Single Hole Bathroom Faucet with Pop-Up Drain Assembly

Check the description for "pop-up drain assembly".

I believe OP is looking for push-pop drain, not pop-up drain - but the general idea is correct :)

To confirm, open the faucet webpage and check the specifications and exploded parts lists.

+1 - look for the manufacturer spec PDF which is sometimes linked in the product description (sometimes webpage descriptions in 3rd party providers like build.com or Home Depot can have mistakes in them) - in Home Depot, for example, the spec PDF is often linked in an Additional Resources box on the right-hand side of the page once your expand the Product Details section:

@bluebird, thank you, that is what I was looking for. I thought there phrase meant the drain was installed without putty etc lol. I will definitely check them out.

@Einav, those are great, and Kohler at that. I swear I look up things, I just don't get the same results. Thank you!

@wdccruise, I wish there was an option for push-pop; unfortunately, including pop-up drain doesn't get me the right kind...

" I wish there was an option for push-pop; unfortunately, including pop-up drain doesn't get me the right kind..."

Sorry, you wrote "but finding one without the metal rod...is difficult" so I thought you wanted a faucet without a metal rod. Faucets with a "pop-up drain assembly" don't have a rod. I see now that you want a faucet with a "clicker drain" but it appears only Kohler uses that term:

• Single-handle Chrome bathroom faucets with clicker drains (all Kohler)

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• Delta single-handle chrome bathroom faucets with various push pop-up drains (browse build.com, Home Depot, etc. for better prices)