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.

irreduciblecharacters over $\mathbb{C}$.) But it may be nontrivial to answer your question, since you allow $p$ to range over all primes dividing the group order, not just the defining characteristic which is usually called $p$ (with $q$ being a power of $p$). In the literature other primes are usually denoted $\ell$ or such. $\endgroup$ – Jim Humphreys Oct 31 '17 at 15:022more comments