Plutarch's 'Parallel Lives' makes note of Spartan customs and society in several places, principally in the 'Life of Lycurgus' and the 'Life of Lysander'. In the former especially he makes notes of how Spartan men were educated and raised. For your question, the following quote:

None of his laws were put into writing by Lycurgus, indeed, one of the so-called "rhetras" [proclamations] forbids it. For he thought that if the most important and binding principles which conduce to the prosperity and virtue of a city were implanted in the habits and training of its citizens, they would remain unchanged and secure, having a stronger bond than compulsion in the fixed purposes imparted to the young by education, which performs the office of a law-giver for every one of them. And as for minor matters, such as business contracts, and cases where the needs vary from time to time, it was better, as he thought, not to hamper them by written constraints or fixed usages, but to suffer them, as occasion demanded, to receive such modifications as educated men determine. Indeed, he assigned the function of law-making wholly and entirely to education. Plutarch's 'Life of Lycurgus' 131-2

I am personally no expert on how long this custom may have lasted in Spartan society. It goes without saying, of course, that ancient scholars rarely listed their own sources and rarely were alive at the same time as the people/events they write about. In Plutarch's case, he was writing over 800 years after Lycurgus is thought to have lived. But I hope this provides something of an answer to your question, at least regarding sources.

Source: Plutarch, 'Parallel Lives', Loeb Classical Library Edition, 1914

Little follow up question, Why were they banned from keeping records?