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We study subsets of $\mathbb{F}_p^n$ that do not contain progressions oflength $k$. We denote by $r_k(\mathbb{F}_p^n)$ the cardinality of such subsetscontaining a maximal number of elements.  In this paper we focus on the case $k=p$ and therefore sets containing nofull line. A trivial lower bound $r_p(\mathbb{F}_p^n)\geq(p-1)^n$ is achievedby a hypercube of side length $p-1$ and it is known that equality holds for$n\in\{1,2\}$. We will however show $r_p(\mathbb{F}_p^n)\geq(p-1)^3+p-2\sqrt{p}$, which is the first improvement in the three dimensionalcase that is increasing in $p$.  We will also give the upper bound $r_p(\mathbb{F}_p^{3})\leqp^3-2p^2-(\sqrt{2}-1)p+2$ as well as generalizations for higher dimensions.  Finally we present some bounds for individual $p$ and $n$, in particular$r_5(\mathbb{F}_5^{3})\geq 70$ and $r_7(\mathbb{F}_7^{3})\geq 225$ which can beused to give the asymptotic lower bound $4.121^n$ for $r_5(\mathbb{F}_5^{n})$and $6.082^n$ for $r_7(\mathbb{F}_7^{n})$.

Maximal line-free sets in $\mathbb{F}_p^n$