The Torex XC6204 LDO regulator (markings '4BMD') is rated for 150 mA at 10 V max. Recommend 6 V ~ 9 V usage.
Updated (information overload ahead):
The 'Pro Micro' is a SparkFun design, as far as I know at least, but the boards you are using are clones. The perils of Open Hardware.
PDF of the SparkFun product, and schematic can be found here:
https://www.sparkfun.com/products/12640#documents-tab
The GitHub:
https://github.com/sparkfun/Pro_Micro
The genuine SparkFun boards use a Microchip MIC5219 regulator in the current revision, with an absolute max of 20 V, and 500 mA. Those parts can be run continuously at 12 V.
I don't remember where (schematic?) but I found a revision note reference to a 400 mA part that was likely in the original design (there were two previous revisions), that has since been upgraded by SparkFun.
Notice that the poly-fuse on the boards is also marked '4' for 400 mA, likely for an older revision board, and way too high for the Torex XC6204!
The regulator on the clone boards are not necessarily Torex XC6204, but may be clones themselves of the Torex part, or a similar related part. The last marking character 'D' is supposed to be the production batch, so I would expect it to change, but it doesn't seem to.
My Google-fu is strong, and I can be relentless. I recently purchased 10 similar boards for under $1 each, and did my research before applying power. I found a listing on an auction site 'yoycart' (new to me) for reels of the Torex XC6204 parts when I searched for 'regulator 4bmd'.
I knew someone would mention the absolute max of the Torex XC6204 part being 12 V, but yes, you don't want to run at that continuously; it's like driving with the accelerator-pedal always on the floor! You must account for power supply ripple, as well as thermal derating margin, especially with an enclosure. Hence the datasheet indicates 10 V max usage, and I recommended 9 V.
Note that the 3.3 V boards should have an 8 MHz crystal, and the 5 V boards a 16 MHz crystal. Running at 16 MHz at 3.3 V is out-of-spec for the ATmega32U4.
