I found a Thomas Gaunt, son of John Gaunt, who was christened at St Michael, Stamford, Lincoln on 6 Nov 1588. Source: England, Select Births and Christenings, 1538-1975, FHL Film Number 504769.
This MIGHT be the same Thomas Gaunt that married Susan Bate on 28 Aug 1609 in Rotherham, Yorkshire. He would have been 21 at the time. They would have had to travel from Rotherham to Lincoln in a year in order to birth Peter in Lincoln in 1610 though.