Let $X$ be a vector space and $E$ be a vector bundle over $X$ where each fiber is a vector space over $F$. Let $\Gamma(E)$ denotes the set of all sections of $E$. Generally $\Gamma(E)$ is infinite dimensional vector space. We want to form a necessary and sufficient conditions of $\Gamma(E)$ is finite dimensional.
Let $X$ be a compact Hausdorff and let $E$ be an $F$ vector bundle over $X$. Then by the result from Swan's paper Vector bundles and projective modules, the set of section denoted by $\Gamma(E)$ is a finite dimensional projective module over $C(X)$. Then there exist a finite subset $\{s_{1},\cdots, s_{n} \}$ which spans $\Gamma(E)$. Thus if $s$ is a section on $E$ then $$ s =\displaystyle \sum_{i=1}^{n} a_{i} s_{i}.$$
Now suppose $X$ be a finite set. Let $X=\{ x_{1},\cdots ,x_{m} \}$. Define $m$ functions $\{ e_{i} \}_{i=1}^{m}$ on $X$ as $ e_{i}(x_{j})=\delta_{ij} $ where $\delta_{ij}$ is Kronecker delta. This collection forms a basis for $C(X)$.
Then any $a_{i} $ in $C(X)$ can be written as $a_{i}= \displaystyle \sum_{j=1}^{m} \lambda_{ij} e_{j}$. Hence we can write a section $s$ as $$ s =\displaystyle \sum_{i=1}^{n} \displaystyle \sum_{j=1}^{m} \lambda_{ij} e_{j} s_{i}.$$
Thus $\Gamma(E)$ is a vector space over $F$ which is spanned by the set $\{e_{j}s_{i} \}_{i,j}$. Hence dimension of $ \Gamma(E)$ is less than or equal to $mn$.
Hence if $X$ is finite compact Hausdorff space then $\Gamma(E)$ is finite dimensional vector space over $F$.
Consider the trivial bundle $E=I \times \mathbb{Z}_{2}$ over $I$ where $I=[0,1]$ and $\mathbb{Z}_{2}=\{0,1\}$ with discrete topology. Consider the section $s$ on $E$. Then $$s(x)=\left( x,f(x) \right)$$ where $f: I \to \mathbb{Z}_{2}$ is continuous, and hence constant.
Hence $\Gamma(E)=\{ s_{1},s_{2} \}$, where $s_{1}(x)=(x,0)$ and $s_{2}(x)=(x,1)$. Thus $\Gamma(E)$ is finite dimensional over the field $\mathbb{Z}_{2}$ where base space $I$ is infinite.
Thus base space is infinite need not imply that $\Gamma(E)$ is infinite dimensional.
But here the underlying field $\mathbb{Z}_{2}$is finite. We want to find a proof for if base space is finite then $\Gamma(E)$ is finite dimensional vector space over a $\mathbb{C}$ or a counter example like the one above.