  Let $V$ be a linear space over a field ${\bf k}$ with a braiding $\tau :
V\otimes V\rightarrow V\otimes V.$ We prove that the braiding $\tau$ has a
unique extension on the free nonassociative algebra ${\bf k}\{V\}$ freely
generated by $V$ so that ${\bf k}\{V\}$ is a braided algebra. Moreover, we
prove that the free braided algebra ${\bf k}\{V\}$ has a natural structure of a
braided nonassociative Hopf algebra such that every element of the space of
generators $V$ is primitive. In the case of involutive braidings, $\tau^2={\rm
id}$, we describe braided analogues of Shestakov-Umirbaev operations and prove
that these operations are primitive operations. We introduce a braided version
of Sabinin algebras and prove that the set of all primitive elements of a
nonassociative $\tau$-algebra is a Sabinin $\tau$-algebra.
