I have a normed space (I'll denote it $C^2[a;b]$) which consists of continuous real functions whose first and second derivatives are also continuous in interval $[a;b]$.
$\forall x,y \in C^2[a;b]$ and $\forall \lambda \in \mathbb{R}$ the norm is defined as follows
$$ \|x(t)\| = |x(a)| + \max_{t \in [a;b]} | x'(t) | + \sqrt{\int_a^b (x''(t))^2dt} ;$$
It is in fact a valid norm following the definition.
But do I have a Banach space? It should follow from definition as well but I have little experience with such formal arguments.
It's not. Start with $z_n(t)$ being continuous, zero for $-1\le t\le0$, one for $1/n\le t\le1$, and linear on $[0,1/n]$. Put $y_n(t)=\int_{-1}^t z_n(\tau)\,d\tau$ and $x_n(t)=\int_{-1}^t y_n(\tau)\,d\tau$.
You now have a Cauchy sequence in your space (on the interval $[-1,1]$ which does not converge. Its limit ought to have been zero for $t<0$ and $\frac12t^2$ for $t\ge0$, and that function is not in your space, since the second derivative is not continuous.