How many Ga atoms are connected to one As atom in solid gallium arsenide?
It is quite well established that stoichiometric gallium arsenide (CAS 1303-00-0) forms a zincblende analogous structure. It consists of two stacked face-centered cubic lattice systems (Ioffe Physical Technical Institute database). See also Uncle Al's Answer. In other words, the arsenic forms the lattice and the gallium fills half of the tetrahedral positions (and vice versa). Therefore every gallium has four arsenic nearest neighbours.
This does not necessarily mean that these are bonds. In this case however, the bonds are best described as covalent (having a somewhat ionic character).
The Britannica Online Encyclopedia (Academic Edition) (You might need access provided through an educational institution) states:

Besides the elemental semiconductors, such as silicon and germanium, some binary crystals are covalently bonded. Gallium has three electrons in the outer shell, while arsenic lacks three. Gallium arsenide (GaAs) could be formed as an insulator by transferring three electrons from gallium to arsenic; however, this does not occur. Instead, the bonding is more covalent, and gallium arsenide is a covalent semiconductor. The outer shells of the gallium atoms contribute three electrons, and those of the arsenic atoms contribute five, providing the eight electrons needed for four covalent bonds. The centres of the bonds are not at the midpoint between the ions but are shifted slightly toward the arsenic.
Such bonding is typical of the III–V semiconductors — i.e., those consisting of one element from the third column of the periodic table and one from the fifth column. Elements from the third column (boron, aluminum, gallium, and indium) contribute three electrons, while the fifth-column elements (nitrogen, phosphorus, arsenic, and antimony) contribute five electrons. All III–V semiconductors are covalently bonded and typically have the zincblende structure with four neighbours per atom. Most common semiconductors favour this arrangement.

This is still a fluorishing field for researchers in the field of materials design semiconductors, semiconductor lasers, computational physics/forcefield development, and more.
The book &quot;Electronic Devices and Circuit Theory&quot; clearly gets it very wrong. I had a look into it and noticed they also argue with the Bohr model, which might not be a very good starting point at all. However, the focus of the book is clearly not solid state chemistry, and therefore it might fit the purpose.