Let $X,Y$ be path-connected finite CW complexes with base points $x_0,y_0$, let $f\colon X\to Y$ be a surjective continuous map, such that for every $y\in Y$, the fiber $f^{-1}(y)$ is path connected. In this case, is the induced map $$f_*\colon\pi_1(X,x_0)\to\pi_1(Y,y_0)$$ on topological fundamental groups necessarily surjective?

(If this is not true in general, will this be true in the case when $Y$ is a smooth complex manifold, $X\subset Y\times\mathbb{P}^n$ a quasi-projective variety, and $f=\mathrm{pr}_1$?)

[I think one sufficient condition is that $f$ satisfies the "arc-lifting property": Sufficiently short arc $(-\epsilon,\epsilon)$ centered at any $y\in Y$ can be lifted to an arc in $X$. For then we can cover a path in $X$ by finitely many arcs in $X$, and join the arcs by path in the fibers. But I am not sure if this is always doable?]