The answer is *yes*.

In fact, it is possible to construct a rational elliptic fibration $f \colon S \to \mathbb{P}^1$ with exactly one multiple fibre of multiplicity $m \geq 2$, by starting from the blow-up of $\mathbb{P}^2$ at nine points that are the base locus of a pencil $\mathscr{P}$ of elliptic curves and then performing a logarithmic transformation centered at one point of $\mathbb{P}^1$.

Since the logarithmic transformation does not change the fibres outside the center, choosing a sufficiently general pencil $\mathscr{P}$ the reduced fibres of $f$ will be all irreducible.

For more details and examples, see

Y. Fujimoto, *On rational elliptic surfaces with multiple fibers*, Publ. Res. Inst. Math. Sci. 26, No.1, 1-13 (1990). ZBL0729.14027,

in particular Proposition 1.1.