No, not by far. The oldest writing systems used for Sanskrit, Brahmi and Kharoshti Script, developed in the 3rd century BC. By that time, there were a number of other writing systems around already (both scripts are actually theorized to be derived from the Phoenician Alphabet, which is as well the origin of the Latin alphabet). What is generally assumed to be the first writing system is Sumerian cuneiform, for which the earliest sources date back until ca. 3300 BC.

As DucklingsOfDoom points out, Sanskrit does not approach the status of our oldest language of which we have attestation of a written tradition. But moreover, I phrase that characterisation precisely as I do because it is even further from being the language of which we have the oldest extant texts.

Sanskrit's claim to fame is really the antiquity within the Indo-European sphere of the composition of the oral tradition of Vedic literature. Outside the Indo-European sphere or in a manuscript context, the antiquity of Vedic literature is not especially remarkable. In the Indo-European sphere, Hittite and Mycenaean Greek, for example, immensely predate it. And outside the European sphere, Egyptian, Sumerian, and the various cuneiform scripts (for unrelated languages) which proceeded from the latter, and alphabetic scripts (mostly Afroasiatic) which proceeded from the prior, do so even more extremely.