There are many examples constructed with weighted shifts. The following is a Hilbert space example.
Let us consider the Hilbert space $L^2\big((0,1),\mu\big)$, where $\mu$ denotes the measure defined by
$$\mu(A):=2\lambda(A\cap(0,\tfrac12))+\lambda(A\cap(\tfrac12,1)).$$
for all Lebesgue measurable sets A. Here $\lambda$
is the Lebesgue-measure. Furthermore, let $T(t)$ be the nilpotent left shift semigroup. Obviously, $T$ satises the semigroup property and, since the norm
$\|\cdot\|_{\mu}$ is equivalent to the norm $\|\cdot\|_{\lambda}$, $T$ is strongly continuous.
Clearly, $\|T(t)\|\leq 2$.
In addition we see that $T(t) = 0$ for all $t>1$.
Finally, consider the function
$$f_t = \frac{1}{\sqrt{t}}\chi_{(\frac12,\frac12+t)}$$
for $t\in(0,\tfrac12)$.
Clearly, $\|f_t\|_{\mu} = 1$ and
$$\|T(t)f_t\|_{\mu} = 2.$$
ADDED:
Theoretically, you can always introduce an equivalent norm in your space which makes your semigroup a contraction semigroup, see Lemma II.3.10 in
Engel, Klaus-Jochen; Nagel, Rainer, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics. 194. Berlin: Springer. xxi, 586 p. (2000). ZBL0952.47036.
The construction is, however, in most cases highly non-constructive and can only used for theoretical purposes.
It is more important that there are examples in Hilbert spaces where it is impossible to find an equivalent Hilbert space norm (i.e., the semigroup is not similar to a contraction semigroup), see
Packel, E. W., A semigroup analogue of Foguel’s counterexample, Proc. Am. Math. Soc. 21, 240-244 (1969). ZBL0175.13802.
