No: there are units that are norms of elements but not norms of units. The simplest example
are real quadratic number fields $Q(\sqrt{m}\,)$ with $m$ a sum of two squares such that the
fundamental unit has positive norm, for example $m = 34$.

For finding other examples, one may look at the ambiguous class number formulas for cyclic extensions $L/K$ of prime degree $p$:
$$ Am(L/K) = h_K \frac{\prod e(P)}{p(E_K : E_K \cap NL^\times)}, \qquad
Am_s(L/K) = h_K \frac{\prod e(P)}{p(E_K : N E_L)}. $$
Here $Am$ denotes the subgroup of ideal classes fixed by the Galois group $G$,
$Am_s$ the subgroup of classes generated by ideals fixed by $G$, and $e(P)$ is the
ramification index of the prime $P$. The index
$$ (Am:Am_s) = (E_K \cap NL^\times : NE_L) $$
is the obstruction to the local-global principle for units.

**Edit.** Let me, however, point out that D. Folk (*When are global units
norms of units?*, Acta Arith. 76 (1996), 145-147) has proved the following
result: if $L/K$ is normal and if $H$ denotes the Hilbert class field of $L$,
then a unit from $K$ that is a local norm in all completions of $H/K$ is the
norm of a unit from $L$. This suggests the following question: given a cyclic
extension $L/K$, is there an unramified abelian extension $E/L$ with the
property that a unit from $K$ is the norm of a unit from $L$ if and only if
it is a local norm in all completions of $E/K$?