Maintainability of writ against private banks

You are right. All the institutions, irrespective of whether they are government controlled or not, can be brought under writ jurisdiction if they perform public functions. There are many judgments of the Supreme Court to this effect.

The Courts in various judgments have held that Writs are maintainable against (i) the State (Government); (ii) an authority; (iii) a statutory body; (iv) an instrumentality or agency of the State; (v) a company which is financed and owned by the State; (vi) a private body run substantially on State funding; (vii) a private body discharging public duty or positive obligation of public nature.
Thus, for your query, if the particular Private Bank is doing anything unlawful, which it shouldn't have as a Public Duty, then only the Writ is maintainable.
Otherwise a private dispute with a Private Bank will not be grounds to hold the Writ maintainable.

Depending upon for what right you are going against the said private bank, writ is maintainable. So clarify what right you want remedyfor through Writ. So give me details so that i may give you a proper answer.