I only happen to know the answer to this question because I had occasion to look up similar things recently. As Jay mentions, asking for a character of this type is the same as asking for a block of $p$-defect zero (which then contains exactly one such character). For finite simple groups of Lie type, the existence of such a block was shown by Michler for all odd primes $p$, and by Willems for $p=2$.
Michler, Gerhard O.
A finite simple group of Lie type has p-blocks with different defects, p≠2.
J. Algebra 104 (1986), no. 2, 220–230.
Willems, Wolfgang. Blocks of defect zero in finite simple groups of Lie type. J. Algebra 113 (1988), no. 2, 511–522.
A complete list of nonabelian finite simple groups $G$ for which there exists a prime $p$ such that $G$ does not have a $p$-block of defect zero can be found in Corollary 2 here:
Granville, Andrew; Ono, Ken.
Defect zero p-blocks for finite simple groups.
Trans. Amer. Math. Soc. 348 (1996), no. 1, 331–347.