A well-known Elizabethan literary critic was raving about an anonymous woman writing around the time.
In 1593, the same year the William Shakespeare's name first appeared in print, a prominent Elizabethan literary critic named Gabriel Harvey was praising an 'excellent gentlewoman'. 
“I dare not Particularise her Description,” he wrote, leaving the object of his praise anonymous; however, he revealed that the woman in question had already written three sonnets and a comedy. Gabriel Harvey wrote of the woman's work, "all her speeches beautified with the grace of Affability … In her mind there appeareth a certain heavenly Logic; in her tongue & pen a divine Rhetoric … I dare undertake with warrant, whatsoever she writeth must needs remain an immortal work, and will leave, in the activest world, an eternal memory of the silliest vermin that she should vouchsafe to grace with her beautiful and allective style, as ingenious as elegant." This sound eerily like the work we have come to associate with the Bard.
There are two issues with assuming that Harvey is talking about Shakespeare when he refers to an 'excellent gentlewoman'. Firstly, at the beginning of his career, Shakespeare was largely unrecognised. In 1593, he was likely still living in London and would not yet have the profile to draw the attention of the country's most prolific literary critics. Secondly, by 1593, Shakespeare had written more than three sonnets and a comedy. He had written five plays himself, co-authored a further three and written two long poems. Therefore, we can assume that Harvey is not talking about Shakespeare when he refers to an 'excellent gentlewoman'.
[P1] Around the time of Shakespeare's emergence, a prominent Elizabethan literary critic was praising an 'excellent gentlewoman' who had written sonnets and a comedy. [P2] This sounds like he was referring to Shakespeare's work. [P3] Therefore, Shakespeare was a woman.
[Rejecting P1] By 1593 Shakespeare had written far more than three sonnets and a comedy. [Rejecting P2] Therefore, Harvey was not referring to Shakespeare.