The Oxford English Dictionary (OED2) lists Sunday best and Sunday's best under the lemma Sunday, but does not provide literary citations for these terms. The oldest instance I could find dates to the late 18th century (my bolding):
The Political Magazine and Parliamentary, Naval, Military, and Literary Journal, For the Year M,DCC,LXXXXI. Volume II.
London: J. Bew 1781, p. 424:
House of Lords
Wednesday May, 30.
Bill for preventing certain abuses and Profanation on the Lord's Day
On the order of the day for committing this Bill.
Lord Abingdon opposed it in the following curious speech.
Sunday being in this country, as in all other Christian countries, the day of otium cum dignitate,
the day of rest, the day when people wash and clean themselves, and as the saying is, put on their
Sunday's best; and their being in this metropolis, some who having washed and cleaned themselves,
and put on their Sunday's best, are willing to enjoy this otium cum dignitate, not by walking
al fresco on a Sunday evening, lest their Sundays best be spoiled by the rain, but under cover [...]
But the concept of a special set of clothes set aside for use on Sundays and religious holidays is certainly older. Searching for Sunday's clothes and Sunday clothes one can trace it back in English-language publications for another century (my bolding):
Jeremiah Burroughs, Four Books On the Eleventh of Matthew, London: Peter Cole 1659, p. 667:
As suppose you have been the last week drunk, or have been committing uncleanness, or have been in Company, and you can come to the Sacrament on the Lord's day, and joyne with the Minister when at prayer, you can come and put on your sunday clothes (as you cal them) and sit at the word and it is as easy to you as any other thing, [...]"
Miguel de Cervantes Saavedra (tr. John Phillips), The History of the Most Renowned Don Quixote of Mancha, and His Trusty Squire, Sancho Pancha, London: Thomas Hodgkin 1687, p. 142:
"Well — at length I discover'd the Cause of my Grief to a young Shepherd
that serv'd my Father: I desir'd him to lend me his Sunday's Clothes,
and to go along with me to the Village where I knew D. Ferdinand was."
According to Wikipedia which cites Samuel Putnam, Phillips's translation is not faithful to the original work. I tracked down the original sentence in the 1605 edition of Don Quixote, and it simply speaks of a shepherd boy's clothing (my bolding):
Que fue, ponerme en este habito, que me dio uno de los que llaman çagales en casa de los labradores, que era criado de mi padre, al qual descubri toda mi defuentura, y le rogue me acompañasse hasta la Ciudad, donde entendi que mi enemigo estava.