Most of Tehillim was written by David about himself. Are there any psalms that are considered prophecy or even messianic prophecy?
Mahartaz Chayus says (Rosh Hashana 4a) that it is perplexing why the Talmud (ibid.) refers to one of the psalm’s authors as prophet. He goes on to say that Tehilim is not said with nevuah only with ruach hakodesh.
Ibn Ezra says that the doxology that closes T'hilim 89 is David Hamelech's reaction to a messianic prophecy:
בָּר֖וּךְ ה' לְ֝עוֹלָ֗ם אָ֘מֵ֥ן וְאָמֵֽן׃
. . . about which he comments
הטעם שראה המשורר ברוח הקודש ביאת המשיח, על כן נתן הודות לשם.
This explanation may arise from a need to identify the motivation for this otherwise incongruously exultive line following a section of scathing despair over adversarial strife.