If the lamp is lit by actual fire, I'd probably say: Put out the lamp.
This is like how you "put out" a barbeque or a camping fire.
Note that, at least where I am, "lamp" and "light" are now synonymous and it took me a while to figure out what difference you were implying between the electric lightbulb and your "lamp".
ell.stackexchange.com/users/626/Lightness Races in Orbit
you can use douse the lamp or extinguish the lamp for a fire-lit lamp.
ell.stackexchange.com/users/57473/Preety Gurditta
A single word for ending fires of all kinds: extinguish : (transitive) to put out, as in fire; to end burning; to quench This works for all kinds of flame lamps equally well (as does the original, put out ): candles:
snuff , snuff out , put out , extinguish gas lights: turn off , put out , extinguish oil lamps: turn off , put out , extinguish burning torches: douse , put out , extinguish "Extinguish" and "put out" also appropriate for electric lights, for which I'd not use any of the other terms above.
ell.stackexchange.com/users/44500/Toby Speight
"Turn out the lamp" was also used more than 'off' in the times when lamps were oil or gas - see this ngram search .
ell.stackexchange.com/users/9024/Pete Kirkham
Snuff or Snuff out applies specifically to lights based on fire: lamps, candles, torches.
It's perhaps a little bit old fashioned (though so are fire-based lights!)
ell.stackexchange.com/users/31107/Beejamin
Your two choices in your scenario are Turn off the lamp.
which is most idiomatic.
Another possibility might be Shut off the lamp.
but less often used.
ell.stackexchange.com/users/26439/Peter
Blow out is an other precise phrase which can be used in this context.
It means: Be extinguished by an air current.
Example usage:
She blew out the candles , posed graciously for countless photographs and accepted cheek kisses galore.
Seattle TimesNov 29, 2016
ell.stackexchange.com/users/56324/Codeformer
We often say turn off the lights. Is it correct to say  turn off the lamp  when referring to a non-electric lamp (lit by fire)?That doesn't sound proper. Which is the better phrase to use there?
What is the correct word for "turn off lamp" for a non-electric lamp?
ell.stackexchange.com
<phrase-usage><usage>
ell.stackexchange.com/questions/133768
ell.stackexchange.com/users/56324/Codeformer
"Put out the lamp" is the term used in this context. If the lamp is lit by actual fire, one probably say:   Put out  the lamp.   
If the lamp is lit by actual fire, one probably say:   Put out  the lamp.   
This is like how you "put out" a barbeque or a camping fire.  
Note that, at least where I am, "lamp" and "light" are now synonymous and it took me a while to figure out what difference you were implying between the electric lightbulb and your "lamp".