Model # BFG1F4040S3N0V RESET BUTTON WON'T STAY LIT

You have a 6 year tank-type low-nox gas water heater
http://www.whirlpoolwaterheaters.com/products/ByCategory.aspx?id=1
Google search 'low nox' for more details about low nox type water heaters that burn fuel so less NO2 irritant is vented into atmosphere.

Owner's manual:
http://www.whirlpoolwaterheaters.com/downloads/6510397.pdf

1) You said 'reset button.' Only mention of 'reset' in manual is about FVIR flammable vapor ignition system where water heater shuts off when exposed to vapors that might catch fire or explode:
This model appears to have Whirlpool's new resettable FVIR so read instructions in manual for resetting FVIR. If FVIR will not reset using instructions, then call whirlpool at 1-877-817-6750

2) You said 'won't stay lit.' Manual has troubleshooting flowchart starting on page 24.
There are 13 items listed for pilot won't stay lit, with possible remedy for each.
There are 9 things for burner will not light, with remedies.
Call whirlpool at 1-877-817-6750

3) Save manual posted above. I install water heaters and tell everyone to read and save manual to empower them to help themselves.

4) Call a plumber for service.

5) Add a comment for more free help
Also take advantage of fixya phone service.
For a price, expert speaks with you over phone while you work on water heater or any do-it-yourself project.
Fixya is always less expensive than a service call.