Tamil. Thampiraan vaNakkam (Goa, India: Henrique Henriques, 1578).
Bengali. Nathaniel Brassey Halhed, A Grammar of the Bengal Language (Hugli, India, 1778).
Hindi. A Grammar of the Hindoostanee Language (Calcutta, India: Chronicle Press, 1796).
Oriya. Mrtyuñjaya Bidyalankar, trans. [New Testament] (Shrirampur, India: Serampore Mission Press, 1807).
Malayalam. [New Testament] (Bombay, India: Courier Press, 1811).
Assamese. William Carey, et al., trans. [New Testament] (Shrirampur, India: Serampore Mission Press, 1813).
Telugu. Grammar of Telugu (Shrirampur, India: Serampore Mission Press, 1813).
As is clear, nearly after a century of the printing of the first English book, we see a Tamil book being printed in Goa (while the other Indian languages were printed nearly two centuries after Tamil). This accidental blessing of the printing press in Goa, and the role of missionaries in setting it up, as well as the Tamil connection is discussed in a recent article by Babu K Verghese in the Hindu:
It was Christian missionaries, who wanted to produce the Bible in the several languages of the country, who introduced printing and publishing in India. In fact, we got the first printing press as a happy accident: As early as 1542, Francis Xavier, a Spaniard, was teaching the Bible in Tharangambadi (Tranquebar), Tamil Nadu. Also, when the Viceroy of Goa, on behalf of King Joan III of Portugal, opened schools for Indians, books had to be provided. Thus, pressure was put on Portugal by Francis Xavier to dispatch printing presses to India, Ethiopia and Japan. Meanwhile, the Emperor of Abyssinia (Ethiopia) requested the king of Portugal to send a press along with the missionaries. Thus the first batch of Jesuit missionaries left for Ethiopia on March 29, 1556. En route, they arrived in Goa on September 6, 1556. But, while they were preparing to proceed to Ethiopia, news reached them that the Ethiopian Emperor was not keen to receive the missionaries. Thus, as luck would have it, the press stayed in Goa and was set up at the College of St. Paul in Goa. Today, the huge arch of the St. Paul’s College gate, restored by the Archaeological Survey of India, stands as a witness to this pioneering effort.
In this regard, I should also mention the Italian priest Veeramamunivar, who compiled several dictionaries and composed literary and grammatical works in Tamil in the early 1700s.
PS: Do not take the dates given above to be the final word on the subject; the author of the posts agrees that some of the dates are educated guesses. So, if you know that the dates are wrong, or if you know of any other Indian language and the year of first printing in the same, leave a note.