  It is well-known that every sharply 2-transitive group of characteristic 3
splits. Here we construct the first examples of non-split sharply 2-transitive
groups in odd positive characteristic $p$, for sufficiently large primes $p$.
Furthermore, we show that any group without 2-torsion can be embedded into a
non-split sharply 2-transitive group of characteristic $p$ for all sufficiently
large primes $p$, yielding $2^{\aleph_0}$ many pairwise non-isomorphic
countable non-split sharply 2-transitive groups in any sufficiently large
characteristic.
