Is it possible that Jacob retained his memories?

Probably - but it's more likely he lost them and regained them later

"In the past it was often the Dark Lord's pleasure to invade the minds of his victims, creating visions designed to torture them into madness."

^{Severus Snape, Harry Potter and the Order of the Phoenix}

Here, Severus Snape is talking about the Legilimency, a form of magic that can best be described as "mind-reading" (a term that Snape hates). From this quote, we can see that Legilimency can also be used to plant thoughts into a persons mind.

Queenie Goldstein, the woman who Jacob smiles at when she enters his bakery at the end of the film, is shown to be unusually powerful in Legilimency. Unlike Snape (himself well-practiced in the art), she is able to read minds at will and with ease.

As such, it is entirely possible that when she returned to the bakery, she "reimplanted" the memories that Jacob has lost, showing him the adventure they had been on and the time they spent together. Although most certainly illegal (the International Statute of Secrecy is taken very seriously), we're shown that she has a reason to do this - she quite clearly disagrees with the idea of Obliviating Jacob, as demonstrated by this dialogue in the film.

Jacob Kowalski: You're not going to Obliviate me?
Queenie: Of course not. You're one of us now.

It's important to note that (at least with the Obliviate spell) the effects can be reversed. In "Harry Potter and the Deathly Hallows", Hermione Granger erases her parents memories before going on the run, intending to restore them when her mission to destroy Voldemort and his Horcrux's is completed. Even if Queenie doesn't use her powerful Legilimency to return his memories to him, there is clearly other ways that the same effect could be achieved.

However, another possibility is that Jacob simply saw a beautiful woman walk into his bakery, who he was naturally attracted to, and smiled. The film makes it pretty clear that Queenie was attracted to him too - this scene could just be a hint that even without his memories fully intact, Queenie intends to form a relationship with him.

I think some memories were retained, as I believe that Queenie erased Jacob's memories (!), not the rain. Thus, Queenie left behind those memories of her. Why do I believe Queenie erased the memories?

1. The rain erases memories of all nomajes both indoors and outdoors, as can be clearly seen in the movie. Thus, if the rain erased Jacob's memory, it would have happened immediately. Even if you say the rain does not work underground, there was a point were they're standing under the glass - almost outdoors - and yet Jacob's memories are not yet erased. Furthermore, while they're standing outside in the rain (!) Jacob's memory is not yet erased. Not until Queenie hugs and kisses him does he lose his memories.

2. The President clearly orders Newt and Tina to obliviate Jacob's memories. If the rain would do it automatically, there would be no need for such an order.

3. The rain erases negative memories. These are positive.

4. If you look closely, you'll see Queenie doing something with her wand, only after which Jacob loses his memories. Again seemingly it looks like its Queenie who erases the memories, no the rain.

Why would Queenie erase Jacob's Memories? Because she wants to marry him. In New York at the time, it was forbidden for nomajes to marry wizards, lest they reveal the magical community. I'm going to assume that this prohibition exists only when the wizard reveals her or himself as such, but should the wizard choose to not practice magic and not reveal herself, it would be permitted. Queenie erased Jacob's memory, except for some choice portions of it (as a legilimens it would probably be easier for her than others to choose which memories to discard and keep) and intends to give up being a witch and thus be allowed to legally marry him without breaking the statute of magical secrecy.

For more on this theory see this link