= Solution
Use the <Weinstein neighborhood theorem>: a neighborhood of a compact <Lagrangian submanifold> is <symplectomorphic> to a neighborhood of the zero section of its <cotangent bundle>, with the identification equal to the identity on that submanifold. Take the canonical sign $-d\lambda$ in this identification. We also use <C1 openness of diffeomorphisms>: on a compact manifold, all smooth self-maps sufficiently close in the $C^1$ topology to a fixed <diffeomorphism> are themselves <diffeomorphisms>.
In $M\times M$ put $\Omega=-\operatorname{pr}_1^*\omega+\operatorname{pr}_2^*\omega$. The diagonal $\Delta$ is <Lagrangian>. For a <symplectomorphism> $\phi$, its graph $\Gamma_\phi$ is also <Lagrangian>, because its pullback of $\Omega$ is $-\omega+\phi^*\omega=0$.
If $\phi$ is sufficiently $C^1$-close to the identity, $\Gamma_\phi$ lies in the fixed Weinstein neighborhood of $\Delta$. Its image in $T^*\Delta$ is transverse to the cotangent fibers and is a section: the projection of that image to $\Delta\cong M$ is $C^1$-close to the identity, hence is a <diffeomorphism> on compact $M$. Reparametrizing by this projection identifies the image with $\operatorname{graph}(\sigma)$ for a small <differential one-form> $\sigma$.
The preceding <graph of a closed one-form is Lagrangian> criterion gives $d\sigma=0$. Since $H^1_{\mathrm{dR}}(M)=0$, the <de Rham cohomology> definition gives $\sigma=df$. Intersections with the zero section are precisely the <critical points> of $f$; under the neighborhood identification these are the intersections $\Gamma_\phi\cap\Delta$, hence the <fixed points> of $\phi$.
On a nonempty compact manifold without boundary, $f$ has a maximum and a minimum. If it is nonconstant, these occur at distinct <critical points>. If it is constant, $df=0$ everywhere, so the whole graph is the diagonal and every point is fixed. \b[For positive-dimensional $M$, there are at least two distinct fixed points.] This is the <nearby exact Lagrangian intersection lemma> applied to the diagonal. No connectedness assumption is needed.
The usual positive-dimensional convention is necessary for the assertion: if zero-dimensional <symplectic manifolds> are allowed, a single-point $M$ has $H^1=0$ and only one <fixed point>. That is a literal exception to the printed statement.
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