Why NH4NO3 has a giant ionic structure?

ionic compound contains unlike charged ions, resulted from electron transfer, binded by electrostatic force(ionic bond), which are arranged in fixed lattice and develops in three dimensions infinitely
eg. NaCl containing Na+ & Cl-
covalent compound contains atoms binded by sharing their respective outermost eletrons (bond pair)
eg. in H2O
Overall, a compound X which contains non-metal constituent elements does NOT imply X is a covalent compound
NH4NO3 consists of cation NH4+ and anion NO3-, therefore, it is an ionic compound
some other compounds, such as NH4+Cl-, O2+[AsF6]-, NO2+ClO4− are also some examples illustrating the fact you have questioned