Existence of rent has nothing to do with whether market is competitive or not. First there are different definitions of rent. As explained in the Palgrave dictionary of economics (Alchian 2017):
Rent’ is the payment for use of a resource, whether it be land, labour, equipment, ideas, or even money. The term is often restricted to payment for use of land or equipment. ‘Economic rent’ is payment for use of any resource whose supply is fixed. Rent serves a social purpose because market levels of rent indicate which uses of fixed resources are the highest valued, and direct such resources to those uses. ‘Monopoly rent’ is paid to producers in markets that are artificially restricted; it may be dissipated by ‘rent seekers’ who compete for monopoly status.
Hence rent can have various meanings depending on the context. Only the last definition of rent that is:
‘Monopoly rent’ is paid to producers in markets that are artificially restricted; it may be dissipated by ‘rent seekers’ who compete for monopoly status.
Has anything to do with competition. Even here there must be some artificial restriction on the market so there is some monopoly (i.e. one seller). All other definitions of rent have nothing to do with competition and some of the above definitions are fully compatible with perfect competition (e.g. even if there would be perfectly competitive market for land and there would be some marginal costs of making it productive - which there actually are although usually they are small (you have to maintain land for it to be usable for housing etc) - the perfectly competitive price for land would still be called rent).
For example, it can trivially be proven that in oligopolistic competition firms can make positive profit, and prices will be higher than marginal costs (see examples Belleflamme & Peitz Industrial organization: markets and strategies). However, oligopolistic profit would not be considered monopoly rent - even if firms have market power.
In order to have monopoly rent in this case you need to make argument that the plot of land is sufficiently distinct from other plots of land so it can be considered to be unique and its owner would thus become a monopolist because there would be only one seller of this one unique piece of land. You can probably make this assumption for any special location (some place with unique excellent view) or locations with some historical value making them special (e.g. land under the White House), but such assumption would be inadequate for any random slot of land in some city. In many cases all plots of land in a given block or even in a whole section of city can be treated as completely homogenous without loss of generality.
