Are there only four demonstrative pronouns this, that, these, and those in English, or are there more like thik which is a variation of this and that?

“Hither, page, and stand by me

        If thou know’st it, telling
Yonder peasant, who is he?

        Where and what his dwelling?”

“Sire, he lives a good league hence

        Underneath the mountain

Right against the forest fence

        By Saint Agnes’ fountain.”

The simple answer is that no, the only standard ones are this, that, these, those.
But I suppose the more nuanced answer depends on the meaning you ascribe to demonstrative, and perhaps even to be.

The OED says demonstratives are:

A word used to indicate the location (spatially, temporally, or abstractly) of something or someone in relation to the discourse context; esp. a demonstrative pronoun or determiner (as, in English, this, that, these, those). Formerly also more generally: †a word used to indicate the person or thing referred to (as, for example, a personal pronoun or a determiner) (obsolete).

• 1768 A. Vieyra New Portuguese Gram. 35 — There are three principal demonstratives in Portuguese, viz. este, this; esse, that; aquelle, that.


• 1833 L. J. A. McHenry Spanish Gram. 42 Possessives and demonstratives are used in Spanish both as adjectives and as pronouns.


• 1875 R. Morris Hist. Eng. Gram. (1877) 114 The Demonstratives are the, that, this, such, so, same, yon.


• 1911 F. Boas Handbk. Amer. Indian Langs. (U.S. Bureau Amer. Ethnol. Bull. No. 40) I. 949 Demonstratives are usually adverbialized by means of another particle.


• 1995 C. E. Schweitzer in F. W. Carové Kinderleben 52 Carové..continues with a phrase in which the initial das is a demonstrative.

The reason the sense of be matters is that whatever others you may find are apt to be one or more of obsolete, archaic, dialectal, or regional in nature, and so of only limited use today.

yon, yond, yonder

The most obvious example is yon, which is a demonstrative determiner for something far away but in sight, and can be roped into use as a demonstrative pronoun for the same. It also has variants yond and yonder, but no matter which way you spell it, it’s no longer in common use throughout the entire anglosphere.

Now archaic and dialect.

A. adj.

1. A demonstrative word used to modify a noun to indicate a thing or person as (literally, or sometimes mentally) pointed out: cf. that adj. 1. Formerly often, as still in some dialects, simply equivalent to that (those); but chiefly, and in later literary use almost always, referring to a visible object at a distance but within view: = ‘that (those)…over there’. Also in yon same.

B. pron. singular or plural.

The adjective used absol., usually denoting a visible object (or objects) pointed out, at a distance but within view: = ‘that (or those) over there’; but sometimes simply = ‘that’ (or ‘those’): cf. A. 1. Now only Scottish and dialect.

To me, it has a rustic feel to it, something from the 19th century or from rural or mountain folk.

thilk

Your other example of thik is less known, at least cisatlantically. It comes out of Middle English and has had several spellings. It can be found in the OED under thilk adj. and pron.

Pronunciation: /ðɪlk/

Etymology: Middle English þilke, known a1300; apparently < þe, the adj., pron.2, and n.1 + ilce, ilk adj.1, pron.1, and n. same, meaning the or that same; in some of the quots. þe ilke or þet ilke occurs as a MS. variant.

This analysis suits the form þilke, but does not explain the early southern þülke and the Kentish þelke, which naturally indicate an Old English *þylce. Can there have been a confusion in the south between þilke and late Old English þylc for þyllic, thellich adj. and pron.?

(Thick /ðɪk/ is in dialect use from Cornwall and Hants to Worcester and Hereford; and also in Pembroke, Glamorgan, and Wexford. In many parts it has also the form thicky, thickee, or thicka. It generally means ‘that’, but in some parts ‘this’, in which case it is contrasted with thuck, thock, or thack = that. It is sometimes indefinite, and has to be made definite, as thick here, this, thick there, that. In Somerset and Dorset, thick and theäse are used only of individual shaped things, as a man or tree, while that and this are used of formless substances in the mass, as flour, milk, marble. See Eng. Dial. Dict.)

archaic or dialect.

A. adj.

a. (determiner). The very (thing, person, etc.) mentioned or indicated; the same; that; this.

†b. With plural n.: These; those. Obsolete.

B. pron.

a. That (or this) person or thing.

†b. plural. Those. Obsolete.

So there are only four demonstrative pronouns in common use today. There are a few other stray forms held over from Middle English that you may find in extremely old writing or in dialect — or in Scotland. :)