As the title suggests, I was wondering if it is possible to play a life domain cleric using the basic rules spell list. I have not bought any additional source material. The reason for my question is that I was viewing the paladin spell list and noticed that it didn't have some essential spells like smite spells and then I noticed that other spellcasters also had some small lists. I understand that this is because they are the basic rules but still wanted to know if I could play cleric with them. Currently I did not see any big problems. Thanks

Seems like it should work. I just scanned the spell list to n basic rules and it looks pretty complete. 
None of the basic rules lists will have every spell a class can have, since many of them appear in supplemental books. 
And paladins hardly ever cast spells anyway. And when they do, it’s almost never one of the smite spells, 99 times out of 100 they’re better off just smiting. 

As far as spells go, yes, very easily. 105 spells are available to the cleric using only the Basic Rules. That’s a lot more than you will ever need. However, I would encourage you to talk to the DM if you have been using spells and find that you would like more. They may permit some homebrew (which is free to make on DDB). Please see my signature for any additional questions about homebrew. 

Thanks for clearing up my doubts.