The term for such sets is "caps". The problem you ask was posed by Bose ("Mathematical theory of the symmetrical factorial design", Sankhyā 8 (1947) 107–166), and is important in relation to coding theory: see Hill, A first course in coding theory (1986), around figure 14.9. At least at the time of Hill's book from which the following values are taken, the maximal size was known just for $p=2$ (it is $2^n$), for $n=2$ (it is $p+1$ for $p$ odd, $p+2$ for $p$ a power of $2$), for $n=3$ (it is $p^2+1$) and in the further cases $(n,p)=(4,3)$ and $(n,p)=(5,3)$ (where it is $20$ and $56$ respectively). (Some of these results are due to Bose himself.)

The term for such sets is "caps". The problem you ask was posed by Bose ("Mathematical theory of the symmetrical factorial design", Sankhyā 8 (1947) 107–166), and is important in relation to coding theory: see Hill, A first course in coding theory (1986), around figure 14.9. At least at the time of Hill's book from which the following values are taken, the maximal size was known just for $p=2$ (it is $2^n$), for $n=2$ (it is $p+1$ for $p$ odd, $p+2$ for $p$ a power of $2$), for $n=3$ (it is $p^2+1$) and in the further cases $(n,p)=(4,3)$ and $(n,p)=(5,3)$ (where it is $20$ and $56$ respectively). (Some of these results are due to Bose himself.)

PS: It might also be worth while to point out that for $n=2$, caps are also known as "arcs", and a theorem of B. Segre states that (for $p$ odd), an arc of maximal size (also, "oval") is necessarily a conic.