No. The narrator of White Nights never says his own name, and no one calls him by a name in the book. He is known by the word with which he introduces himself to Nastenka on the First Night: dreamer. We know that he is twenty-six, that he has lived for eight years in St. Petersburg and that in all that time he has made not a single acquaintance.
The lack of a name matches the way he sees himself. On the First Night he says: "I am a dreamer." On the Second, when Nastenka wants to know who he is, he answers that he is a type, and goes on to describe the Petersburg dreamer in the third person, as a species and not as a person, because in the first person he is ashamed. He himself prefers to be a specimen rather than someone.
The book's other names reinforce the contrast. Nastenka says hers only on the Second Night, and he declares that he will never tire of repeating it. The lodger also has no name and is called only that to the end; the grandmother is the grandmother. Proper names are kept by the women who surround the two of them: Nastenka, Matryona, his maid, and Fiekla, the grandmother's maid.
Some read the anonymity as a device so that any reader can put himself in his place, and others read it as part of the portrait: a man who lives by reverie and has left no mark on anyone does not need a name, because no one would call him. The text sustains both readings.