Is the lamprey,
Pteromyzon (Fig. 1), the basalmost fish? Or is it a large, derived lancelet (Fig. 2) with eyes, external gill openings and parasagittal fins?

The answer might come down to the question,
is a skull present in lampreys? Fish have a skull (Fig. 4). Lancelets (Fig. 2) do not.

Cambrian lancelets
like Metaspriggina (Fig. 3) have eyes, but no skull. So do flatworms, lampreys and conodont animals. Transitional taxa between flatworms and lampreys, like nematodes and extant lancelets, lack eyes. So do all the closest relatives of basal chordates: echinoderms, crinoids, tunicates. The loss of eyes appears to be a derived trait in hagfish (Fig. 3) and, by convergence, chordate relatives derived from lancelets. Lamprey eyes are complex structures that took a long time to evolve. Apparently blindness is a derived trait on generally sessile chordate cousins, starting with lancelets.

Perhaps now is a good time to consider definitions
to help decide the lamprey ‘fish or lancelet’ question.

According to Wikipedia (Lamprey),
“Lampreys are an ancient extant lineage of jawless fish of the order Petromyzontiformes, placed in the superclass Cyclostomata.

According to Wikipedia (Fish),
“Fish are gill-bearing aquatic craniate animals that lack limbs with digits.”

Note that both definitions rely on traits,
not phylogenetic placement.

Even so,
does the lamprey skull (= cranial and annular cartilage) have homologs in the dermal skull of craniates?

Not really, except perhaps, the nasal,
which is the largest bone/cartilage in many basal fish. The appearance of more dermal cranial bones would have been phylogenetically gradual. We wait until Drepanaspis, Drepanolepis and Birkenia for cranial homologs of many bones to appear in these valid jawless craniates.

You might think the loss of eyes in lancelets
resulted in the loss of primitive cartilage structures that held the eyes, as in large-eyed conodonts, but even in lampreys, such dermal skull structures are not present (Fig. 1).

It’s always phylogeny, not a short list of defining traits.
In the large reptile tree (LRT, 1729+ taxa; subset Fig. 5) the last common ancestor of lampreys (genus: Pteromyzon; Fig. 1), is the lancelet Metaspriggina (Fig. 3) The last common ancestor of all extant fish is Euphanerops, another jawless fish from the Late Devonian with an earlier radiation in the Silurian. Middle Silurian Birkenia (Fig. 4) is the most primitive fish with skull bones homologous with those of extant fish. A less primitive, Early Silurian Jamoytius (Fig. 5) has been considered a sister to lampreys, but in the LRT nests between Birkenia and Thelodus.

Armored lancelets
Arandaspis, and Porapsis have an armored gill chamber with few to no bone homologs with those of living craniates. Yes, that armor forms a sort of skull, but it is not the same skull as in Birkenia and its craniate descendants. Arandaspis, and Porapsis have been called primitive fish, but phylogenetically precede valid craniates, which means they cannot be ‘fish’, by definition. For the same reason, neither can the lamprey, Pteromyzon, be called a fish.

Lamprey origins
go back to the early Ordovician. 38 extant and 5 extinct species are known. Like other lancelets and basal fish, primitive and young lampreys are filter feeders. The closest living relatives of lampreys are hagfish (Fig. 2) and lancelets.

As hypothesized earlier
here, the lack of a lens, iris and eyeball in lancelets was retained by derived taxa including tunicates and echinoderms, distinct from the order proposed by Romer and others.

Another point-of-view, according to Gregory 1933:
“The profound researches of Stensio (1927) and Kiser (1924) have left no reasonable doubt however, that one or another of the ostracoderms gave rise to the modern class of cyclostomes, including the lampreys and hags, thus confirming the earlier views of Cope and others.”

Gregory 1933 continues:
“Both Gaskell and Stensio have shown the striking similarities of the larval lamprey head to that of the cephalaspid ostracoderms and in the light of much evidence it seems highly probable that the lamprey skull type has been derived from a cephalaspid-like type in the following way:

1. “the bony exoskeleton has lost its bone cells and become membranous; 
2. thorny epidermal teeth have developed around the sucker-like mouth;
3. a rasping apparatus has developed out of the so-called tongue, which is a specialized part of the branchial apparatus;
4. the rest of the branchial arches have been displaced backward;
5. the cartilages that support the sucker and its teeth have also been enlarged; 
6. the originally continuous cartilaginous septa between the gill-pouches have become fenestrated, giving rise to the branchial basket;
7. a special hydraulic organ, described by T. E. Reynolds (1931) has been developed in the oral chamber to assist in the sucking action of the mouth.”

LRT arguments to Gregory’s 1933
ostracoderm ancestry for lampreys based on the LRT:

1. Gregory has it backwards: the bony exoskeleton had not developed yet;
2. Agreed. Lamprey ‘teeth’ are not homologous with those of gnathostomes;
3. Or the tongue is a new medial organ not associated with paired lateral branchials;
4. Not displaced relative to LRT sisters;
5. Agreed;
6. Or the fenestrations are derived from sisters to Metaspriggina, unknown in 1933;
7. Agreed, but only a few lampreys are blood suckers and no juveniles are bloodsuckers.

This is not the first time the LRT has been in the minority.
In the LRT(subset Fig. 5) the lamprey is several nodes more primitive than the ostracoderm, Hemicyclaspis, a sturgeon ancestor with pectoral fins, armor and a heterocercal tail. Again, the best way to find out what a taxon is, is to nest it in a wide gamut cladogram like the LRT.

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References
Gregory WK 1933. Fish skulls. A study of the evolution of natural mechanisms. American Philosophical Society 23(2) 1–481.
Linnaeus C 1758. Systema naturæ per regna tria naturæ, secundum classes, ordines, genera, species, cum characteribus, differentiis, synonymis, locis. Tomus I. Editio decima, reformata.

wiki/Branchiostoma
wiki/Lamprey_Pteromyzon
wiki/Birkenia
wiki/Jamoytius